text: CLASH Cluster Morphologies 21 Table 3 X-ray Morphological Properties for the CLASH Sample 500 kpc Name RA Dec C C w w P30 P30 P40 P40 AR AR PA PA centroid centroid unc unc unc unc unc unc A209 22.97016 -13.6118 0.167 0.005 5.2E-03 1.0E-03 4.8E-08 5.6E-08 3.78E-08 3.3E-08 0.91 0.01 -49 3 A383 42.01374 -3.52945 0.525 0.004 1.8E-03 4.7E-04 5.8E-08 2.7E-08 6.03E-09 6.1E-09 0.97 0.01 22 11 0329-02 52.42264 -2.19577 0.488 0.006 1.1E-02 1.1E-03 1.6E-07 8.5E-08 6.76E-08 3.8E-08 0.95 0.02 -31 6 0416-24 64.03614 -24.07162 0.091 0.004 1.5E-02 4.0E-03 1.5E-06 7.4E-07 8.54E-07 3.5E-07 0.75 0.02 37 2 0429-02 67.39994 -2.88486 0.531 0.006 4.8E-03 1.4E-03 2.0E-07 1.0E-07 2.67E-08 2.0E-08 0.89 0.02 -14 4 0647+70 101.95787 70.24891 0.242 0.008 1.0E-02 2.3E-03 6.4E-07 3.1E-07 5.68E-07 2.8E-07 0.76 0.02 -80 2 0717+37 109.38513 37.75319 0.055 0.004 4.5E-02 3.4E-03 2.5E-06 9.7E-07 9.75E-07 4.1E-07 0.84 0.02 -29 2 0744+39 116.21812 39.45748 0.365 0.006 2.4E-02 1.6E-03 1.4E-07 1.2E-07 2.81E-07 1.0E-07 0.98 0.01 -22 37 A611 120.23689 36.05725 0.335 0.004 2.0E-03 4.6E-04 8.0E-08 4.0E-08 2.33E-08 1.6E-08 0.95 0.01 33 4 1115+01 168.96666 1.49813 0.430 0.008 5.6E-03 1.7E-03 1.9E-07 1.2E-07 1.68E-08 2.5E-08 0.85 0.03 -29 3 1149+22 177.39769 22.4004 0.111 0.007 1.9E-02 3.2E-03 5.5E-06 1.6E-06 5.27E-07 3.3E-07 0.81 0.02 -36 2 A1423 179.32236 33.61042 0.287 0.004 4.6E-03 7.0E-04 6.8E-08 5.4E-08 4.13E-08 3.0E-08 0.91 0.02 55 3 1206-08 181.55339 -8.8027 0.223 0.007 3.6E-02 2.0E-03 2.3E-06 4.4E-07 5.29E-07 1.4E-07 0.85 0.01 10 2 1226+33 186.74102 33.54674 0.245 0.010 5.5E-03 2.1E-03 1.9E-07 2.5E-07 6.05E-08 7.4E-08 0.95 0.03 -69 13 1311-03 197.75704 -3.17733 0.488 0.013 3.2E-03 1.2E-03 1.0E-07 1.1E-07 1.10E-07 7.4E-08 0.89 0.04 -16 6 1347-1145 206.87852 -11.753 0.506 0.003 1.4E-02 3.6E-04 1.3E-07 2.5E-08 1.48E-08 7.2E-09 0.84 0.01 -1 1 1423+24 215.94964 24.07839 0.555 0.009 3.5E-03 1.4E-03 1.6E-07 1.2E-07 6.20E-08 4.9E-08 0.88 0.02 17 7 1532+30 233.22438 30.34978 0.571 0.007 1.9E-03 8.3E-04 1.2E-08 1.3E-08 2.64E-08 1.8E-08 0.92 0.02 55 6 1720+35 260.06941 35.60649 0.417 0.008 6.1E-03 1.3E-03 2.8E-08 3.7E-08 2.52E-08 2.3E-08 0.93 0.02 11 6 A2261 260.61273 32.13266 0.331 0.004 4.7E-03 5.7E-04 1.2E-07 4.2E-08 1.25E-08 9.4E-09 0.93 0.01 63 3 1931-26 292.95663 -26.57594 0.545 0.006 1.7E-03 6.2E-04 3.5E-08 2.8E-08 6.56E-08 2.3E-08 0.80 0.01 -4 2 2129-07 322.3573 -7.69189 0.211 0.010 2.7E-02 2.8E-03 7.0E-07 4.0E-07 4.85E-08 5.6E-08 0.90 0.02 84 6 2129+0005 322.41548 0.08858 0.426 0.004 3.8E-03 5.7E-04 6.1E-08 2.8E-08 2.45E-08 1.2E-08 0.87 0.01 70 2 MS2137 325.0635 -23.66098 0.589 0.007 2.5E-03 7.9E-04 5.1E-08 5.7E-08 5.60E-08 3.7E-08 0.92 0.03 66 8 2248-44 342.18683 -44.52922 0.194 0.003 1.5E-02 9.7E-04 2.0E-07 7.1E-08 2.74E-08 1.6E-08 0.91 0.01 62 2 Note. ∼— The uncertainty range reported in this table and Table 4 is computed from the diﬀerence between the upper and lower 68 percent ( 1σ) of the ﬁts after 100 Monte Carlo runs, and the reported morphology value is the median value from those runs. AR is axis ratio; PA is Position Angle, in degrees east of north. Table 4 X-ray Morphological Properties for the CLASH Sample 0.5R500 Name RA Dec C C w w P30 P30 P40 P40 AR AR PA PA centroid centroid unc unc unc unc unc unc A209 22.96919 -13.6111 0.215 0.005 5.0E-03 1.0E-03 2.8E-08 3.0E-08 1.9E-08 1.8E-08 0.88 0.02 -46.2 2.5 A383 42.01382 -3.52943 0.594 0.004 1.6E-03 4.1E-04 5.2E-09 5.5E-09 1.4E-08 8.2E-09 0.98 0.01 -11.5 19.4 0329-02 5·2·.·42257 -2··.1·957 0·.·5·11 0·.0··06 1·.2··E-02 1·.·1·E-03 1·.5··E-07 6·.5··E-08 2·.·4·E-08 1·.·8·E-08 0·.9··1 0·.0··2 -2··8·.3 5·.0·· 0416-24 ··· 0429-02 6·7·.·40009 -2··.8·8479 0·.5··63 ··· 4·.7··E-03 1·.·1·E-03 4·.·4·E-08 4·.6··E-08 9·.·3·E-09 1·.3··E-08 0·.8··9 0·.·0·2 -1··1·.9 4·.7·· 0647+70 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· 0717+37 0744+39 116.21804 39.45748 0.375 0.007 2.3E-02 1.6E-03 1.5E-07 1.1E-07 2.0E-07 8.2E-08 0.98 0.01 -18.9 30.7 A611 120.23706 36.05736 0.394 0.004 2.4E-03 7.3E-04 7.9E-09 9.4E-09 4.6E-09 4.8E-09 0.94 0.01 40.7 5.1 1115+01 1·6·8·.96671 1·.·4·9809 0·.4··81 0·.0··08 5·.3··E-03 1·.8··E-03 8·.8··E-08 7·.7··E-08 1·.·0·E-08 1·.2··E-08 0·.8··7 0·.·0·2 -·2·7·.4 5·.1·· 1149+22 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· A1423 1206-08 181.55361 -8.80286 0.262 0.006 3.2E-02 1.4E-03 1.5E-06 3.2E-07 2.0E-07 7.2E-08 0.85 0.01 9.0 2.2 1226+33 186.74112 33.54693 0.315 0.011 6.8E-03 1.6E-03 4.0E-07 3.0E-07 6.9E-08 5.9E-08 0.93 0.02 -86.3 10.5 1311-03 197.75701 -3.17731 0.477 0.012 3.9E-03 1.1E-03 7.9E-08 1.0E-07 1.0E-07 9.1E-08 0.86 0.03 -14.4 5.8 1347-1145 206.87867 -11.75312 0.585 0.003 1.2E-02 3.6E-04 5.1E-08 1.4E-08 1.4E-08 4.8E-09 0.85 0.01 0.4 1.2 1423+24 215.94964 24.07835 0.562 0.009 4.7E-03 1.4E-03 6.9E-08 7.1E-08 3.1E-08 3.3E-08 0.89 0.03 18.0 7.0 1532+30 233.22436 30.34977 0.587 0.007 2.0E-03 7.9E-04 2.0E-08 2.0-E-08 7.7E-09 8.4E-09 0.93 0.02 51.7 9.0 1720+35 260.06927 35.60642 0.458 0.007 5.5E-03 1.2E-03 5.4E-08 4.9E-08 2.0E-08 2.0E-08 0.94 0.02 22.4 5.2 A2261 260.6115 32.13223 0.431 0.004 5.2E-03 5.0E-04 5.7E-07 8.5E-08 2.1E-07 3.4E-08 0.86 0.01 73.9 1.4 1931-26 2·9·2·.95663 -·2·6·.57591 0·.·5·75 0·.0··05 1·.8··E-03 5·.·9·E-04 4·.0··E-08 2·.7··E-08 2·.·4·E-08 1·.·6·E-08 0·.7··9 0·.·0·1 -·4·.2· 1·.8·· 2129-07 2129+00 322.41523 0.08861 0.461 0.004 4.8E-03 8.1E-04 3.9E-08 2.2E-08 3.2E-08 1.4E-08 0.87 0.01 68.8 1.9 MS2137 325.06373 -23.66072 0.612 0.007 5.4E-03 9.9E-04 3.1E-08 3.4E-08 2.5E-08 2.2E-08 0.97 0.03 60.3 7.3 2248-44 342.18634 -44.52951 0.333 0.003 2.2E-02 7.1E-04 1.8E-08 1.3E-08 5.2E-09 4.7E-09 0.91 0.01 55.2 2.3 Note. — Same convention as previous Table. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CLASH

text: Cluster

text: Morphologies

text: Table

text: X-ray

text: Morphological

text: Properties

text: for

text: the

text: CLASH

text: Sample

text: kpc

text: Name

text: RA

text: Dec

text: C

text: C

text: w

text: w

text: AR

text: AR

text: PA

text: PA

text: centroid

text: centroid

text: unc

text: unc

text: unc

text: unc

text: unc

text: unc

text: Note.

text: The

text: uncertainty

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text: reported

text: in

text: this

text: table

text: and

text: Table

text: is

text: computed

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text: the

text: diﬀerence

text: between

text: the

text: upper

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text: percent

text: of

text: the

text: ﬁts

text: after

text: Monte

text: Carlo

text: runs,

text: and

text: the

text: reported

text: morphology

text: value

text: is

text: the

text: median

text: value

text: from

text: those

text: runs.

text: AR

text: is

text: axis

text: ratio;

text: PA

text: is

text: Position

text: Angle,

text: in

text: degrees

text: east

text: of

text: north.

text: Table

text: X-ray

text: Morphological

text: Properties

text: for

text: the

text: CLASH

text: Sample

text: Name

text: RA

text: Dec

text: C

text: C

text: w

text: w

text: AR

text: AR

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text: PA

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text: unc

text: unc

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text: 5·.0··

text: 4·.7··

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text: Note.

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text: convention

text: as

text: previous

text: Table.

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text: CLASH Cluster Morphologies 5 Table 1 CLASH Clusters and Chandra X-ray Observations Cluster Name RA Dec z 0.5R500 Chandra (—) (hh:mm:ss) (dd:mm:ss) (—) (arcsec) Obs ID Abell 209 01:31:52.54 -13:36:40.4 0.206 203 3579, 522 Abell 383 02:48:03.40 -03:31:44.9 0.187 219 2321 MACSJ0329-02 03:29:41.56 -02:11:46.1 0.450 1·0··1 6108, 3582 MACSJ0416-24 04:16:08.38 -24:04:20.8 0.396 10446 MACSJ0429-02 04:29:36.05 -02:53:06.1 0.399 1·1··3 3271 MACSJ0647+70 06:47:50.27 +70:14:55.0 0.584 ··· 3584, 3196 MACSJ0717+37 07:17:32.63 +37:44:59.7 0.548 4200 MACSJ0744+39 07:44:52.82 +39:27:26.9 0.686 74 6111 Abell 611 08:00:56.82 +36:03:23.6 0.288 149 3194 MACSJ1115+01 11:15:51.90 +01:29:55.1 0.355 1·2··4 9375 MACSJ1149+22 11:49:35.69 +22:23:54.6 0.544 ··· 3589, 1656 Abell 1423 11:57:17.36 +33:36:37.47 0.213 11724 MACSJ1206-08 12:06:12.09 -08:48:04.4 0.439 109 3277 CLJ1226+3332 12:26:58.25 +33:32:48.6 0.890 81 5014, 3180, 932 MACSJ1311-03 13:11:01.80 -03:10:39.8 0.494 81 6110, 9381 RXJ1347-1145 13:47:30.62 -11:45:09.4 0.451 117 3592 MACSJ1423+24 14:23:47.88 +24:04:42.5 0.545 82 4195 MACSJ1532+30 15:32:53.78 +30:20:59.4 0.363 105 1665, 1649 MACSJ1720+35 17:20:16.78 +35:36:26.5 0.387 115 6107 Abell 2261 17:22:27.18 +32:07:57.3 0.224 217 5007 MACSJ1931-26 19:31:49.62 -26:34:32.9 0.352 1·1··7 3282, 9382 MACSJ2129-07 21:29:26.06 -07:41:28.8 0.570 3595, 3199 RXJ2129+0005 21:29:39.96 +00:05:21.2 0.234 161 552, 9370 MS2137-2353 21:40:15.17 -23:39:40.2 0.313 148 4974, 5250 RXJ2248-4431 22:48:43.96 -44:31:51.3 0.348 141 4966 Note. — List of CLASH clusters with their RA, Dec, and redshifts from Postman et al. (2012b). The MACSJ0416-24 cluster redshift has been updated since the Postman et al. (2012) work based on VLT and SOAR spectroscopy. The angular quantity 0.5R500 is computed from lensing-based M500 masses derived in Merten et al. (2015), in arcseconds, assuming the cosmology assumed in that work. − Mpc; n = 0.95 as power spectrum index; and h = 0.7 ticles. The gravitational softening was set to 6h 1 kpc. as reduced Hubble parameter (h = 1). The CLASH Ten diﬀerent simulation snapshots were stored. We ana- 70 sample is comprised of clusters with gas temperatures lyze those four that cover the same redshift range of the − above 5 keV. Assuming M T relations from Vikhlinin CLASH sample: z = 0.25, 0.33, 0.43, 0.67, respectively. et al. (2009), this temperature limit corresponds to the a 3.4.1. Mock X-ray Catalog × − mass limit equal to M > 3.5 1014 M(cid:12)h 1, which is ∼ 500 For each simulated halo, we produce three Chandra- satisﬁed by 100 clusters at z = 0.333 (only 1 simulated − like event ﬁles corresponding to the orthogonal line-of- cluster has mass M > 1015 M(cid:12)h 1 at that redshift). 500 sight projections of the original cosmological volume. The re-simulations involved Lagrangian regions of 6 − These projections are therefore randomly oriented rela- h 1 Mpc radius around the most massive halos (with − tive to a cluster. The three ﬁles are not co-added and are virial masses above 1015h 1M(cid:12) at redshift z = 0) and analyzed independently. The tool adopted is the X-ray were performed by using the TREEPM+SPH code GAD- MAp Simulator (X-MAS,Gardini et al. 2004; Rasia et al. GET (Springel 2005). Two sets of re-simulations were 2008) which accounts for the ancillary response function produced including both non-radiative and radiative and the redistribution matrix function of the ACIS-S3 physics. In this work, as in Meneghetti et al. (2014a), detector on board of the Chandra satellite. The ﬁeld we focus on the non-radiative simulation because the ra- of view is set equal to 16 arcmin, which corresponds to diative run did not include any prescription for feedback ∼ 4.5 Mpc at z = 0.333. The two outer radii used in the by active galactic nuclei, implying that the cluster core current analysis, 500 kpc and 0.5R , are amply within is aﬀected by over-cooling (Borgani & Kravtsov 2011). 500 the mock X-ray image. To generate the event ﬁles we as- In the radiative case, the condensation of X-ray lumi- sume a ﬁxed metallicity with value equal to 0.3 solar (as nous gas in the center is extreme (Rasia et al. 2013a), tabulated by Anders & Grevesse 1989); we include the the light concentration is not realistic, and the cluster × galactic absorption with a WABS model (N = 5 1020 isophotal shapes are less in agreement with observations − H cm 2); and we impose an exposure time equal to 100 ks. than non-radiative simulations (Lau et al. 2012). We As we have done for the real X-ray observations, to note that in a comparison project of various codes and evaluate the morphological parameters from the mock AGN-feedback implementation schemes (Sembolini et al. catalogue, we produce soft X-ray band ([0.5-2] keV) im- 2015), that most of the simulations that include AGN × feedback give similar dark matter distributions and gas ages binned by 2 2 arcsec2. For a detailed description fractions as those found in the non-radiative simulations of the method, see Rasia et al. 2013c. outside of cluster cores. 3.4.2. Mock SZE catalogue The mass resolution of the MUSIC-2 simulations corre- × − To evaluate the eﬀect of the Bolocam PSF on the mor- sponds to m = 9.01 108h 1M(cid:12) for the dark-matter DM × − phological parameters (see Section 4.2), we produce maps particles and m = 1.9 108h 1M(cid:12) for the gas par- gas of the Compton y parameter. For each simulated cluster, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CLASH

label: paragraph

text: Cluster

text: Morphologies

text: Table

text: CLASH

text: Clusters

text: and

text: Chandra

text: X-ray

text: Observations

text: Cluster

text: Name

text: RA

text: Dec

text: z

text: Chandra

text: (hh:mm:ss)

text: (dd:mm:ss)

text: (arcsec)

text: Obs

text: ID

text: Abell

text: Abell

text: 1·0··1

text: 1·1··3

text: Abell

text: 1·2··4

text: Abell

text: Abell

text: 1·1··7

text: Note.

text: List

text: of

text: CLASH

text: clusters

text: with

text: their

text: RA,

text: Dec,

text: and

text: redshifts

text: from

text: Postman

text: et

text: al.

text: The

text: cluster

text: redshift

text: has

text: been

text: updated

text: since

text: the

text: Postman

text: et

text: al.

text: work

text: based

text: on

text: VLT

text: and

text: SOAR

text: spectroscopy.

text: The

text: angular

text: quantity

text: is

text: computed

text: from

text: lensing-based

text: masses

text: derived

text: in

text: Merten

text: et

text: al.

text: in

text: arcseconds,

text: assuming

text: the

text: cosmology

text: assumed

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text: that

text: work.

text: Mpc;

text: n

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text: power

text: spectrum

text: index;

text: and

text: h

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text: The

text: gravitational

text: softening

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text: set

text: to

text: kpc.

text: as

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text: Hubble

text: parameter

text: (h

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text: CLASH

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text: were

text: stored.

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text: comprised

text: of

text: clusters

text: with

text: gas

text: temperatures

text: lyze

text: those

text: four

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text: redshift

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text: the

text: above

text: keV.

text: Assuming

text: M

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text: Vikhlinin

text: CLASH

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text: z

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text: et

text: al.

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text: temperature

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text: Mock

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text: Catalog

text: mass

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text: M

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text: For

text: each

text: simulated

text: halo,

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text: satisﬁed

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text: z

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text: redshift

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text: tool

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text: and

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text: redistribution

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text: the

text: physics.

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text: et

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text: detector

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text: of

text: the

text: Chandra

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text: The

text: ﬁeld

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text: the

text: non-radiative

text: simulation

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text: ra-

text: of

text: view

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text: set

text: equal

text: to

text: arcmin,

text: which

text: corresponds

text: to

text: diative

text: run

text: did

text: not

text: include

text: any

text: prescription

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text: feedback

text: Mpc

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text: z

text: The

text: two

text: outer

text: radii

text: used

text: in

text: the

text: by

text: active

text: galactic

text: nuclei,

text: implying

text: that

text: the

text: cluster

text: core

text: current

text: analysis,

text: kpc

text: and

text: are

text: amply

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text: is

text: aﬀected

text: by

text: over-cooling

text: (Borgani

text: Kravtsov

text: the

text: mock

text: X-ray

text: image.

text: To

text: generate

text: the

text: event

text: ﬁles

text: we

text: as-

text: In

text: the

text: radiative

text: case,

text: the

text: condensation

text: of

text: X-ray

text: lumi-

text: sume

text: a

text: ﬁxed

text: metallicity

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text: value

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text: solar

text: (as

text: nous

text: gas

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text: the

text: center

text: is

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text: [12] L. E. Ibanez, V. M. Lozano, arXiv:1512.08777; E. Ma, arXiv:1512.09159; L. Mar- zola, et al., arXiv:1512.09136; Y. Jiang, Y.-Y. Li, T. Liu, arXiv:1512.09127; A. E. C. Hernandez, arXiv:1512.09092; S. Kanemura, N. Machida, S. Odori, T. Shindou, arXiv:1512.09053; S. Kanemura, et al., arXiv:1512.09048; X.-J. Huang, W.-H. Zhang, Y.- F. Zhou, arXiv:1512.08992; Y. Hamada, T. Noumi, S. Sun, G. Shiu, arXiv:1512.08984; S. K. Kang, J. Song, arXiv:1512.08963; C.-W. Chiang, M. Ibe, T. T. Yanagida, arXiv:1512.08895; A. Dasgupta, M. Mitra, D. Borah, arXiv:1512.09202. [13] A. E. Faraggi, J. Rizos, arXiv:1601.03604; A. Djouadi, J. Ellis, R. Godbole, J. Quevil- lon, arXiv:1601.03696; J. H. Davis, M. Fairbairn, J. Heal, P. Tunney, arXiv:1601.03153; R. Ding, Z.-L. Han, Y. Liao, X.-D. Ma, arXiv:1601.02714; M. Fabbrichesi, A. Urbano, arXiv:1601.02447; J. Cao, et al., arXiv:1601.02570; P. Ko, T. Nomura, arXiv:1601.02490; S. Fichet, G. v. Gersdorﬀ, C. Royon, arXiv:1601.01712; I. Sahin, arXiv:1601.01676; D. Borah, S. Patra, S. Sahoo, arXiv:1601.01828; S. Bhattacharya, S. Patra, N. Sahoo, N. Sahu, arXiv:1601.01569; F. DEramo, J. de Vries, P. Panci, arXiv:1601.01571; H. Ito, T. Moroi, Y. Takaesu, arXiv:1601.01144; A. E. C. Hernandez, I. d. M. Varzielas, E. Schu- macher, arXiv:1601.00661; T. Modak, S. Sadhukhan, R. Srivastava, arXiv:1601.00836; C. Csaki, J. Hubisz, S. Lombardo, J. Terning, arXiv:1601.00638; U. Danielsson, R. Enberg, G. Ingelman, T. Mandal, arXiv:1601.00624; D. Palle, arXiv:1601.00618; K. Ghorbani, H. Ghorbani, arXiv:1601.00602; X-F. Han, et al., arXiv:1601.00534; E. Palti, arXiv:1601.00285; P. Ko, Y. Omura, C. Yu, arXiv:1601.00586; T. Nomura, H. Okada, arXiv:1601.00386; H. Zhang, arXiv:1601.01355; S. Jung, J. Song, Y. W. Yoon, arXiv:1601.00006; I. Dorsner, S. Fajfer, N. Kosnik, arXiv:1601.03267; C. Hati, arXiv:1601.02457; D. Stolarski, R. V.- Morales, arXiv:1601.02004; A. Berlin, arXiv:1601.01381; F. F. Deppisch, C. Hati, S. Patra, P. Pritimita, U. Sarkar, arXiv:1601.00952; B. Dutta, et al., arXiv:1601.00866; A. Karozas, S. F. King, G. K. Leontaris, A. K. Meadowcroft, arXiv:1601.00640; W. Chao, arXiv:1601.00633; arXiv:1601.04678; C. T. Potter, arXiv:1601.00240; A. Ghoshal, arXiv:1601.04291; T. Nomura, H. Okada, arXiv:1601.04516. [14] CMS Collaboration, arXiv:1502.07400. [15] ATLAS Collaboration, arXiv:1508.03372. [16] A. Crivellin, G. DAmbrosio, J. Heeck, Phys. Rev. Lett. 114, 151801 (2015); Y. Omura, E. Senaha, K. Tobe, JHEP 1505, 028 (2015); S. P. Das, J. H.-Sanchez, A. Rosado and R. 15

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text: H.-Sanchez,

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text: Rosado

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text: R.


text: The muon g-2 can be also corrected by the two-loop Barr-Zee diagrams with the fermions loops, W and Goldstone loops. Using the well-known classical formulates (cid:88) [31], the main contributions of two-loop Barr-Zee diagrams in this model are given as − αm (cid:20) δa = µ (cid:88) Nc Q2 y y F (x ) µ2 4π3m f f φµµ φff φ fφ f φ=h,H,A,P;f=t,b,τ,T,Li αm 23 µ (cid:21) + y g 3F (x ) + F (x ) 8π3v φµµ φWW H Wφ 4 A Wφ φ=h,H 3 m2 { − } + G (x ) + φ F (x ) F (x ) , (28) 4 Wφ 2m2 H Wφ A Wφ W where x = m2/m2, x = m2 /m2, g = s , g = c and fφ f φ Wφ W φ HWW α hWW α (cid:90) − − − y (cid:90) 1 1 2x(1 x) x(1 x) F (y) = F (y) = dx − − log (for φ = h, H) (29) φ H 2 x(1 x) y y 0 − (cid:90) y 1 1(cid:20) x(1 x) (cid:21) F (y) = F (y) = dx − − log (for φ = A, P ) (30) φ A 2 x(1 x) y y 0 − −y 1 1 − y x(1 x) G(y) = dx − − 1 − − log . (31) 2 x(1 x) y x(1 x) y y 0 The experimental value of muon g-2 excess is [32] ± × − δa = (26.2 8.5) 10 10. (32) µ → (7) The data of Br(τ µγ). The LFV coupling of the Higgs boson gives the dominant → → contributions to the decay τ µγ. The branching ratio of τ µγ is given by → | | | | BR(τ µγ) 48π3α ( A + A + A 2 + A + A + A 2) → = 1L0 1Lc 2L 1R0 1Rc 2R , (33) BR(τ µν¯ ν ) G2 µ τ F where A , A , A and A are from the one-loop diagrams with the Higgs bosons 1L0 1Lc 1R0 1Rc (cid:20) (cid:18) (cid:19) (cid:21) (cid:88) and tau lepton [17], and ∗ y ∗ m2 − 3 y A = φ τµ y log φ + φ ττ , (34) 1L0 16π2m2 φ ττ m2 2 6 φ=h, H, A, P φ τ † − (ρ(cid:0)e ρe)µτ (cid:1) A = , (35) 1Lc 192π2m2 − ∗ H→ ↔ ∗ A = A y y , y y , (36) 1R0 1L0 φ τµ φ µτ φ ττ φ ττ A = 0. (37) 1Rc 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

label: list

text: muon

text: can

text: be

text: also

text: corrected

text: by

text: the

text: two-loop

text: Barr-Zee

text: diagrams

text: with

text: the

text: fermions

text: loops,

text: W

text: and

text: Goldstone

text: loops.

text: Using

text: the

text: well-known

text: classical

text: formulates

text: the

text: main

text: contributions

text: of

text: two-loop

text: Barr-Zee

text: diagrams

text: in

text: this

text: model

text: are

text: given

text: as

text: αm

text: δa

text: µ

text: Nc

text: y

text: y

text: F

text: (x

text: µ2

text: f

text: f

text: φµµ

text: φff

text: φ

text: fφ

text: f

text: φ=h,H,A,P;f=t,b,τ,T,Li

text: αm

text: µ

text: y

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text: (x

text: F

text: (x

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text: Wφ

text: φ=h,H

text: G

text: (x

text: φ

text: F

text: (x

text: F

text: (x

text: Wφ

text: H

text: Wφ

text: A

text: Wφ

text: W

text: where

text: x

text: x

text: g

text: s

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text: and

text: fφ

text: f

text: φ

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text: φ

text: HWW

text: α

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text: y

text: x)

text: x)

text: F

text: (y)

text: F

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text: dx

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text: φ

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text: H)

text: φ

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text: x)

text: y

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text: y

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text: F

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text: F

text: (y)

text: dx

text: log

text: (for

text: φ

text: A,

text: P

text: φ

text: A

text: x)

text: y

text: y

text: −y

text: y

text: x)

text: G(y)

text: dx

text: log

text: x)

text: y

text: x)

text: y

text: y

text: The

text: experimental

text: value

text: of

text: muon

text: excess

text: is

text: δa

text: µ

text: The

text: data

text: of

text: Br(τ

text: µγ).

text: The

text: LFV

text: coupling

text: of

text: the

text: Higgs

text: boson

text: gives

text: the

text: dominant

text: contributions

text: to

text: the

text: decay

text: τ

text: µγ.

text: The

text: branching

text: ratio

text: of

text: τ

text: µγ

text: is

text: given

text: by

text: BR(τ

text: µγ)

text: A

text: A

text: A

text: A

text: A

text: A

text: BR(τ

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text: τ

text: F

text: where

text: A

text: A

text: A

text: and

text: A

text: are

text: from

text: the

text: one-loop

text: diagrams

text: with

text: the

text: Higgs

text: bosons

text: and

text: tau

text: lepton

text: and

text: y

text: y

text: A

text: φ

text: τµ

text: y

text: log

text: φ

text: φ

text: ττ

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text: ττ

text: φ=h,

text: H,

text: A,

text: P

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text: ρe)µτ

text: A

text: H→

text: A

text: A

text: y

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text: A

text: ##LTLine##

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text: FIG. 1: Under the constrains of the oblique parameters and the LHC Higgs data, the surviving samples projected on the planes of κ versus s , ρ versus m , Y versus κ and ρ versus s . (cid:96) α µτ H L (cid:96) µτ α The circles (green) are allowed by the muon g-2, the pluses (red) allowed by the muon g-2 and → → → Br(τ µγ), and the bullets (black) allowed by the muon g-2, Br(τ µγ) and Br(h µτ). FIG. 2: Under the constrains of the oblique parameters of electroweak data, the LHC Higgs data, → → muon g-2, BR(τ µγ) and Br(h µτ), the surviving samples projected on the planes of Γ P versus s , Y versus Γ and Y versus s . Here 2 fb < R < 4 fb for the circles (green), 4 fb α L P L θ γγ < R < 6 fb for the pluses (red), and 6 fb < R < 10 fb for the bullets (black), with R denoting γγ γγ γγ the 750 GeV Higgs production rate at the 13 TeV LHC. correlation between s and κ due to the experimental constraints of the τ¯τ channel data of α (cid:96) 125 GeV Higgs. The surviving samples have two diﬀerent 125 GeV Higgs couplings to τ¯τ, and their absolute values are around the SM value. One is the SM-like Higgs coupling with the same sign as the coupling of the gauge boson, and the other is the Yukawa coupling with the opposite sign to the coupling of the gauge boson for a relative large κ . (cid:96) From the upper-right panel of Fig. 1, we see that the muon g-2 favors ρ to increase µτ with m . As shown in Eq. (27), the muon g-2 can obtain positive contributions from the H H loop and negative contributions from A and P loops for ρ = ρ . With the decreasing µτ τµ of the mass splitting of H and A, the cancelation between the contributions of H and A loops becomes sizable so that a large ρ is required to enhance the muon g-2. From the µτ → lower-left panel, we see that the upper bound of τ µγ favors a large absolute value of κ (cid:96) → for a large Y . The vector-like leptons with a large Y can sizably enhance Br(τ µγ) via L L the two-loop Barr-Zee diagrams, and such contributions can be partially canceled by the one-loop diagram for a properly large κ and a proper sign of s . The lower-right panel shows (cid:96) θ → that the experimental data of Br(h µτ) requires ρ to increase with the decreasing of µτ the absolute value of s , and -0.05 < s < 0.05 and 0.05 < ρ < 0.7 are favored by the α α µτ FIG. 3: Under the constrains of the oblique parameters of electroweak data, the LHC Higgs data, → → muon g-2, Br(τ µγ) and Br(h µτ), the surviving samples with 20 GeV < Γ < 40 GeV P projected on the planes of the widths of the main decay modes of the 750 GeV Higgs versus λ , m s H → → → and λ . Here Γ(P hA) for the circles (green), Γ(P HZ) for the triangles (blue), Γ(P HA) 7 → ± ∓ for the pluses (red) and Γ(P W H ) for the bullets (black). 10

text: FIG.

text: Under

label: caption

text: the

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text: the

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text: FIG.

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text: fb

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text: the

text: bullets

text: (black),

text: with

text: R

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text: TeV

text: LHC.

text: correlation

text: between

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text: τ¯τ

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text: the

text: muon

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text: A

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text: becomes

text: sizable

text: so

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text: ρ

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text: µτ

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text: panel,

text: we

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text: µγ)

text: via

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text: the

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text: by

text: the

text: one-loop

text: diagram

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text: κ

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text: proper

text: sign

text: of

text: s

text: The

text: lower-right

text: panel

text: shows

text: θ

text: that

text: the

text: experimental

text: data

text: of

text: Br(h

text: µτ)

text: requires

text: ρ

text: to

text: increase

text: with

text: the

text: decreasing

text: of

text: µτ

text: the

text: absolute

text: value

text: of

text: s

text: and

text: s

text: and

text: ρ

text: are

text: favored

text: by

text: the

text: α

text: α

text: µτ

text: FIG.

text: Under

text: the

text: constrains

text: of

text: the

text: oblique

text: parameters

text: of

text: electroweak

text: data,

text: the

text: LHC

text: Higgs

text: data,

text: muon

text: Br(τ

text: µγ)

text: and

text: Br(h

text: µτ),

text: the

text: surviving

text: samples

text: with

text: GeV

text: Γ

text: GeV

text: P

text: projected

text: on

text: the

text: planes

text: of

text: the

text: widths

text: of

text: the

text: main

text: decay

text: modes

text: of

text: the

text: GeV

text: Higgs

text: versus

text: λ

text: m

text: s

text: H

text: and

text: λ

text: Here

text: Γ(P

text: hA)

text: for

text: the

text: circles

text: (green),

text: Γ(P

text: HZ)

text: for

text: the

text: triangles

text: (blue),

text: Γ(P

text: HA)

text: for

text: the

text: pluses

text: (red)

text: and

text: Γ(P

text: W

text: H

text: for

text: the

text: bullets

text: (black).


text: 12 0.3 ε = 0.3 0.2 ε = 0.226 0.1 ε = 0.1 0 0 0.1 0.2 0.3 x−ECC Fig. 5. Density evolution for joint CAC-ECC decoding using a regular (3, 12) LDPC code. 1.00 1.00 0.75 0.75 0.50 0.50 0.25 0.25 0 0 0 0.226 1 0 0.226 1 ε ε (a) Probability of bit error P . (b) Probability of block error P . b e Fig. 6. Monte-Carlo simulations of the probabilities of bit and block error for a regular (3, 12) LDPC code. The plots are for block lengths ∈ { } of N 102, 103, 104, 105 (from light gray to black). A phase transition at the analytically derived threshold of 0.226 is apparent. It is worth commenting on the behavior of the curves around erasure rate ε = 0. The error probability is not zero in this regime due to insufﬁcient free wires being treated as error events. For N = 102 and R = 0.8 as used here, a Binomial approximation estimates this error event as happening with ECC ≈ probability 0.149. The simulation results for N = 102 yield P (0) 0.141. As the block length increases, e ♦ this error event has exponentially decreasing probability and is no longer apparent on the plots. APPENDIX A PROOF OF THEOREM 1 Inform. K L CAC bits Bus a P M ECC Shielding Fig. 7. Shielded joint CAC-ECC encoder from [10]. bP + CCEx P e ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: ε

text: ε

text: ε

text: x−ECC

text: Fig.

text: Density

text: evolution

text: for

text: joint

text: CAC-ECC

text: decoding

text: using

text: a

text: regular

text: LDPC

text: code.

text: ε

text: ε

text: (a)

text: Probability

text: of

text: bit

text: error

text: P

text: (b)

text: Probability

text: of

text: block

text: error

text: P

text: b

text: e

text: Fig.

text: Monte-Carlo

text: simulations

text: of

text: the

text: probabilities

text: of

text: bit

text: and

text: block

text: error

text: for

text: a

text: regular

text: LDPC

text: code.

text: The

text: plots

text: are

text: for

text: block

text: lengths

text: of

text: N

text: (from

text: light

text: gray

text: to

text: black).

text: A

text: phase

text: transition

text: at

text: the

text: analytically

text: derived

text: threshold

text: of

text: is

text: apparent.

text: It

text: is

text: worth

text: commenting

text: on

text: the

text: behavior

text: of

text: the

text: curves

text: around

text: erasure

text: rate

text: ε

text: The

text: error

text: probability

text: is

text: not

text: zero

text: in

text: this

text: regime

text: due

text: to

text: insufﬁcient

text: free

text: wires

text: being

text: treated

text: as

text: error

text: events.

text: For

text: N

text: and

text: R

text: as

text: used

text: here,

text: a

text: Binomial

text: approximation

text: estimates

text: this

text: error

text: event

text: as

text: happening

text: with

text: ECC

text: probability

text: The

text: simulation

text: results

text: for

text: N

text: yield

text: P

text: As

text: the

text: block

text: length

text: increases,

text: e

text: this

text: error

text: event

text: has

text: exponentially

text: decreasing

text: probability

text: and

text: is

text: no

text: longer

text: apparent

text: on

text: the

text: plots.

text: APPENDIX

text: A

text: PROOF

text: OF

text: THEOREM

text: Inform.

text: K

text: L

text: CAC

text: bits

text: Bus

text: a

text: P

text: M

text: ECC

text: Shielding

text: Fig.

text: Shielded

text: joint

text: CAC-ECC

text: encoder

text: from

text: bP

text: CCEx

text: P

text: e

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: 18 (cid:88) (cid:88) Substituting this expression and (7) into (6) yields that ∞ − ρ˜ d d y = 1 p (x ). CAC d,i CAC d d=2 i=1 This concludes the proof. REFERENCES [1] G. D. Micheli and L. Benini, “Networks on chips: a new SoC paradigm,” Computer, vol. 35, pp. 70–78, Jan. 2002. [2] G. D. Micheli, C. Seiculescu, S. Murali, L. Benini, F. Angiolini, and A. Pullini, “Networks on chips: From research to products,” in Proc. ACM/IEEE DAC, pp. 300–305, June 2010. [3] N. R. Shanbhag, “Reliable and efﬁcient system-on-chip design,” Computer, vol. 37, pp. 42–50, Mar. 2004. [4] R. Ho, K. W. Mai, and M. A. Horowitz, “The future of wires,” Proc. IEEE, vol. 89, pp. 490–504, Apr. 2001. [5] P. P. Sotiriadis, Interconnect modeling and optimization in deep submicron technologies. PhD thesis, MIT, 2002. [6] B. Victor and K. Kreutzer, “Bus encoding to prevent crosstalk delay,” Proc. IEEE/ACM ICCAD, pp. 57–63, Nov. 2001. [7] T. K. Konstantakopoulos, “Implementation of delay and power reduction in deep sub-micron buses using coding,” Master’s thesis, MIT, 2002. [8] C.-S. Changa, J. Cheng, T.-K. Huang, X.-C. Huang, and D.-S. Lee, “A bit-stufﬁng algorithm for crosstalk avoidance in high speed switching,” in Proc. IEEE INFOCOMM, pp. 1–9, Mar. 2010. [9] “DDR4 SDRAM standard.” JEDEC, 2013. [10] S. R. Sridhara and N. R. Shanbhag, “Coding for system-on-chip networks: A uniﬁed framework,” IEEE Trans. VLSI Syst., vol. 13, pp. 655–667, June 2005. [11] S. R. Sridhara and N. R. Shanbhag, “Coding for reliable on-chip buses: A class of fundamental bounds and practical codes,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 26, pp. 977–982, May 2007. [12] H. Jin, A. Khandekar, and R. McEliece, “Irregular repeat-accumulate codes,” Proc. 2nd Int. Symp. Turbo Codes, pp. 1–8, Sept. 2000. [13] T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge University Press, 2008. [14] H.-A. Loeliger, “An introduction to factor graphs,” IEEE Signal Process. Mag., vol. 21, pp. 28–41, Jan. 2004. [15] P. P. Sotiriadis and A. Chandrakasan, “A bus energy model for deep submicron technology,” IEEE Trans. VLSI Syst., vol. 10, pp. 341– 350, June 2002. [16] P. P. Sotiriadis and A. Chandrakasan, “Reducing bus delay in submicron technology using coding,” in Proc. IEEE ASP-DAC, pp. 109– 114, Feb. 2001. [17] T. Richardson and R. Urbanke, “The capacity of low-density parity check codes under message-passing decoding,” IEEE Trans. Inform. Theory, vol. 47, pp. 599–618, Feb. 2001. [18] T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inform. Theory, vol. 47, pp. 619–637, Feb. 2001. ##LTLine##

text: Substituting

text: this

text: expression

text: and

text: into

text: yields

text: that

text: ρ˜

text: d

text: d

text: y

text: p

text: (x

text: CAC

text: d,i

text: CAC

text: d

text: This

text: concludes

text: the

text: proof.

text: REFERENCES

text: G.

text: D.

text: Micheli

text: and

text: L.

text: Benini,

text: “Networks

text: on

text: chips:

text: a

text: new

text: SoC

text: paradigm,”

text: Computer,

text: vol.

text: pp.

text: Jan.

text: G.

text: D.

text: Micheli,

text: C.

text: Seiculescu,

text: S.

text: Murali,

text: L.

text: Benini,

text: F.

text: Angiolini,

text: and

text: A.

text: Pullini,

text: “Networks

text: on

text: chips:

text: From

text: research

text: to

text: products,”

text: in

text: Proc.

text: ACM/IEEE

text: DAC,

text: pp.

text: June

text: N.

text: R.

text: Shanbhag,

text: “Reliable

text: and

text: efﬁcient

text: system-on-chip

text: design,”

text: Computer,

text: vol.

text: pp.

text: Mar.

text: R.

text: Ho,

text: K.

text: W.

text: Mai,

text: and

text: M.

text: A.

text: Horowitz,

text: “The

text: future

text: of

text: wires,”

text: Proc.

text: IEEE,

text: vol.

text: pp.

text: Apr.

text: P.

text: P.

text: Sotiriadis,

text: Interconnect

text: modeling

text: and

text: optimization

text: in

text: deep

text: submicron

text: technologies.

text: PhD

text: thesis,

text: MIT,

text: B.

text: Victor

text: and

text: K.

text: Kreutzer,

text: “Bus

text: encoding

text: to

text: prevent

text: crosstalk

text: delay,”

text: Proc.

text: IEEE/ACM

text: ICCAD,

text: pp.

text: Nov.

text: T.

text: K.

text: Konstantakopoulos,

text: “Implementation

text: of

text: delay

text: and

text: power

text: reduction

text: in

text: deep

text: sub-micron

text: buses

text: using

text: coding,”

text: Master’s

text: thesis,

text: MIT,

text: C.-S.

text: Changa,

text: J.

text: Cheng,

text: T.-K.

text: Huang,

text: X.-C.

text: Huang,

text: and

text: D.-S.

text: Lee,

text: “A

text: bit-stufﬁng

text: algorithm

text: for

text: crosstalk

text: avoidance

text: in

text: high

text: speed

text: switching,”

text: in

text: Proc.

text: IEEE

text: INFOCOMM,

text: pp.

text: Mar.

text: SDRAM

text: standard.”

text: JEDEC,

text: S.

text: R.

text: Sridhara

text: and

text: N.

text: R.

text: Shanbhag,

text: “Coding

text: for

text: system-on-chip

text: networks:

text: A

text: uniﬁed

text: framework,”

text: IEEE

text: Trans.

text: VLSI

text: Syst.,

text: vol.

text: pp.

text: June

text: S.

text: R.

text: Sridhara

text: and

text: N.

text: R.

text: Shanbhag,

text: “Coding

text: for

text: reliable

text: on-chip

text: buses:

text: A

text: class

text: of

text: fundamental

text: bounds

text: and

text: practical

text: codes,”

text: IEEE

text: Trans.

text: Comput.-Aided

text: Design

text: Integr.

text: Circuits

text: Syst.,

text: vol.

text: pp.

text: May

text: H.

text: Jin,

text: A.

text: Khandekar,

text: and

text: R.

text: McEliece,

text: “Irregular

text: repeat-accumulate

text: codes,”

text: Proc.

text: Int.

text: Symp.

text: Turbo

text: Codes,

text: pp.

text: Sept.

text: T.

text: Richardson

text: and

text: R.

text: Urbanke,

text: Modern

text: Coding

text: Theory.

text: Cambridge

text: University

text: Press,

text: H.-A.

text: Loeliger,

text: “An

text: introduction

text: to

text: factor

text: graphs,”

text: IEEE

text: Signal

text: Process.

text: Mag.,

text: vol.

text: pp.

text: Jan.

text: P.

text: P.

text: Sotiriadis

text: and

text: A.

text: Chandrakasan,

text: “A

text: bus

text: energy

text: model

text: for

text: deep

text: submicron

text: technology,”

text: IEEE

text: Trans.

text: VLSI

text: Syst.,

text: vol.

text: pp.

text: June

text: P.

text: P.

text: Sotiriadis

text: and

text: A.

text: Chandrakasan,

text: “Reducing

text: bus

text: delay

text: in

text: submicron

text: technology

text: using

text: coding,”

text: in

text: Proc.

text: IEEE

text: ASP-DAC,

text: pp.

text: Feb.

text: T.

text: Richardson

text: and

text: R.

text: Urbanke,

text: “The

text: capacity

text: of

text: low-density

text: parity

text: check

text: codes

text: under

text: message-passing

text: decoding,”

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: vol.

text: pp.

text: Feb.

text: T.

text: Richardson,

text: A.

text: Shokrollahi,

text: and

text: R.

text: Urbanke,

text: “Design

text: of

text: capacity-approaching

text: irregular

text: low-density

text: parity-check

text: codes,”

text: IEEE

text: Trans.

text: Inform.

text: Theory,

text: vol.

text: pp.

text: Feb.

text: ##LTLine##


text: 9 { } at random from 0, 1 N has the same asymptotic distribution of alternating runs. This also shows that, if R > 3/4, then there are asymptotically sufﬁcient number of free wires to hold all parities. ECC Note that so far we have only described the length of the alternating runs of a, but not their actual { } values. These are chosen by selecting the ﬁrst bit of a uniformly over 0, 1 independently of everything else. The remaining bits of a are then fully speciﬁed by the alternating run structure. We consider an ECC that is randomly selected from an ensemble of IRA codes as described next. The information bits are ﬁrst encoded using the CAC on the wires in b corresponding to the past bus state a . For the purpose of analysis, we assume that the CAC-encoded information bits are chosen uniformly 1 C at random from the collection of all CAC codewords (a ) compatible with a . These bits are then CAC 1 1 fed to the IRA code as systematic bits. Then, the free wires in b corresponding to the past bus state a 2 are set to the parity bits of the IRA using the code’s parity-accumulate structure. We start with the description of the connection of CAC check nodes and information variable nodes. Unlike what is depicted in Fig. 3, we represent each alternating run in a by a single CAC check node 1 of degree d. This CAC check node is connected to the corresponding d information variable nodes in b. We continue with the description of the accumulate structure of the IRA. We generate (cid:96)(a ) ECC 2 checks, one for each of the parity variable nodes. We pair the ECC check nodes and the parity variable nodes by connecting them with an edge. Note that the number of ECC checks is a random quantity here, chosen to match the random realization of (cid:96)(a ). We further add an accumulation structure on the parity 2 bits as shown in Fig. 3. This accumulation structure is needed to ensure that the code has a nontrivial threshold. We next describe the construction of the LDPC code linking the (cid:96)(a ) information variable nodes with 1 the (cid:96)(a ) ECC check nodes. The design rate of this LDPC code is 2 − (cid:96)(a ) 2 R = 1 LDPC (cid:96)(a ) −1 ≈ − 1 R ECC 1 R ECC − 1 = 2 , (3) R ECC where the second equality follows from (2) and where the approximation is due to the concentration → ∞ ∈ of (cid:96)(a ) and (cid:96)(a ) around their expected values as N . Note that, since R (3/4, 1], this design 1 2 ∈ ECC rate satisﬁes R (2/3, 1]. LDPC Denote by λ(x) and ρ(x) the edge-perspective degree distributions of the LDPC code. Denote by (L(x), R(x)) the normalized variable and check degree distribution generator polynomials from a node perspective corresponding to the normalized degree distribution generator polynomials from an edge perspective (λ(x), ρ(x)). The degree distributions are chosen to satisfy the desired design rate, (cid:48) − 1 − L (1) 2 = RLDPC = 1 (cid:48) R R (1) ECC or, more succinctly, (cid:48) L (1) 1 − (cid:48) = 1. R (1) R ECC The LDPC code linking the ECC checks with the information variable nodes is now generated by randomly connecting the information variable nodes to the ECC check nodes as in the standard LDPC ensemble using the degree distributions (L(x), R(x)). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: at

text: random

text: from

text: N

text: has

text: the

text: same

text: asymptotic

text: distribution

text: of

text: alternating

text: runs.

text: This

text: also

text: shows

text: that,

text: if

text: R

text: then

text: there

text: are

text: asymptotically

text: sufﬁcient

text: number

text: of

text: free

text: wires

text: to

text: hold

text: all

text: parities.

text: ECC

text: Note

text: that

text: so

text: far

text: we

text: have

text: only

text: described

text: the

text: length

text: of

text: the

text: alternating

text: runs

text: of

text: a,

text: but

text: not

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text: a

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text: threshold.

text: We

text: next

text: describe

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text: construction

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text: The

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text: degree

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text: LDPC

text: code.

text: Denote

text: by

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text: R(x))

text: the

text: normalized

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text: edge

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text: (λ(x),

text: ρ(x)).

text: The

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text: are

text: chosen

text: to

text: satisfy

text: the

text: desired

text: design

text: rate,

text: L

text: RLDPC

text: R

text: R

text: ECC

text: or,

text: more

text: succinctly,

text: L

text: R

text: R

text: ECC

text: The

text: LDPC

text: code

text: linking

text: the

text: ECC

text: checks

text: with

text: the

text: information

text: variable

text: nodes

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text: now

text: generated

text: by

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text: information

text: variable

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text: ECC

text: check

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text: standard

text: LDPC

text: ensemble

text: using

text: the

text: degree

text: distributions

text: (L(x),

text: R(x)).

text: ##LTLine##

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text: 13 (cid:48) · · · (cid:48) C(Λ/Λ , σ2) = C(Λ/Λ , σ2) + + C(Λ − /Λ , σ2). (28) 1 r 1 The key idea of the AWGN-good polar lattices is to use a good component polar code to achieve the capacity C(Λ − /Λ , σ2) for each level (cid:96) = 1, 2, . . . , r. A polar lattice L is resulted from those component polar codes. For (cid:96) 1 (cid:96) such a construction, the total decoding error probability with multi-stage decoding is bounded by (cid:88) (cid:0) (cid:1) ≤ r C (cid:48) P (L, σ2) P ( , σ2) + P (Λ )N, σ2 , (29) e e (cid:96) e (cid:96)=1 (cid:0) (cid:1) C C → where P ( , σ2) denotes the decoding error probability of polar code at level (cid:96). To make P (L, σ2) 0, e (cid:96) (cid:96) e (cid:48) (cid:48) → we need to choose the bottom lattice Λ such that the uncoded error probability P (Λ )N, σ2 0 and construct e C C a code (cid:96) for each Λ(cid:96)−1/Λ(cid:96) channel such that the decoding error probability Pe( (cid:96), σ2) also tends (cid:16)to zero(cid:17). Note that the mod-Λ channel is not used for communication and C(Λ, σ2) is required to be negligible. → To sum up, in order to approach the Poltyrev capacity of AWGN channels, we would like to have log γL(σ) 0 2πe → while P (L, σ2) 0. According to the analysis in [26], we have the following three design criteria: e • The top lattice Λ gives negligible capacity C(Λ, σ2). (cid:48) (cid:48) • The bottom lattice Λ has a small error probability P (Λ , σ2). e C • Each component polar code is a capacity-approaching code for the Λ − /Λ channel. (cid:96) (cid:96) 1 (cid:96) Since polar codes are capacity-achieving, polar lattices are proved to be AWGN-good for a properly chosen lattice (cid:48) partition [29]. The concepts of mod-Λ channel, Λ/Λ channel and AWGN-goodness will be generalized to fading channels in the next subsection. B. Polar Lattices for Fading channels Without Power Constraint For i.i.d. fading channels, the channel gain varies. The above analysis for AWGN channels need to be generalized. Since the receiver knows the CSI, the fading effect can be removed by multiplying Y with 1 . We deﬁne the fading H mod-Λ channel as follows. modfading.eps Fig. 7. A block diagram of the fading mod-Λ channel. V Deﬁnition 4 (fading mod-Λ channel): A fading mod-Λ channel is a fading channel with an input in (Λ), and V an output being scaled by 1 before the mod- (Λ) operation. A block diagram of this model is shown in Fig. 7. H May 28, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: C(Λ/Λ

text: C(Λ/Λ

text: C(Λ

text: /Λ

text: r

text: The

text: key

text: idea

text: of

text: the

text: AWGN-good

text: polar

text: lattices

text: is

text: to

text: use

text: a

text: good

text: component

text: polar

text: code

text: to

text: achieve

text: the

text: capacity

text: C(Λ

text: /Λ

text: for

text: each

text: level

text: r.

text: A

text: polar

text: lattice

text: L

text: is

text: resulted

text: from

text: those

text: component

text: polar

text: codes.

text: For

text: such

text: a

text: construction,

text: the

text: total

text: decoding

text: error

text: probability

text: with

text: multi-stage

text: decoding

text: is

text: bounded

text: by

text: r

text: C

text: P

text: (L,

text: P

text: P

text: (Λ

text: )N,

text: σ2

text: e

text: e

text: e

text: C

text: C

text: where

text: P

text: denotes

text: the

text: decoding

text: error

text: probability

text: of

text: polar

text: code

text: at

text: level

text: To

text: make

text: P

text: (L,

text: e

text: e

text: we

text: need

text: to

text: choose

text: the

text: bottom

text: lattice

text: Λ

text: such

text: that

text: the

text: uncoded

text: error

text: probability

text: P

text: (Λ

text: )N,

text: σ2

text: and

text: construct

text: e

text: C

text: C

text: a

text: code

text: for

text: each

text: channel

text: such

text: that

text: the

text: decoding

text: error

text: probability

text: Pe(

text: also

text: tends

text: Note

text: that

text: the

text: mod-Λ

text: channel

text: is

text: not

text: used

text: for

text: communication

text: and

text: C(Λ,

text: is

text: required

text: to

text: be

text: negligible.

text: To

text: sum

text: up,

text: in

text: order

text: to

text: approach

text: the

text: Poltyrev

text: capacity

text: of

text: AWGN

text: channels,

text: we

text: would

text: like

text: to

text: have

text: log

text: γL(σ)

text: while

text: P

text: (L,

text: According

text: to

text: the

text: analysis

text: in

text: we

text: have

text: the

text: following

text: three

text: design

text: criteria:

text: e

text: The

text: top

text: lattice

text: Λ

text: gives

text: negligible

text: capacity

text: C(Λ,

text: The

text: bottom

text: lattice

text: Λ

text: has

text: a

text: small

text: error

text: probability

text: P

text: (Λ

text: e

text: C

text: Each

text: component

text: polar

text: code

text: is

text: a

text: capacity-approaching

text: code

text: for

text: the

text: Λ

text: /Λ

text: channel.

text: Since

text: polar

text: codes

text: are

text: capacity-achieving,

text: polar

text: lattices

text: are

text: proved

text: to

text: be

text: AWGN-good

text: for

text: a

text: properly

text: chosen

text: lattice

text: partition

text: The

text: concepts

text: of

text: mod-Λ

text: channel,

text: Λ/Λ

text: channel

text: and

text: AWGN-goodness

text: will

text: be

text: generalized

text: to

text: fading

text: channels

text: in

text: the

text: next

text: subsection.

text: B.

text: Polar

text: Lattices

text: for

text: Fading

text: channels

text: Without

text: Power

text: Constraint

text: For

text: i.i.d.

text: fading

text: channels,

text: the

text: channel

text: gain

text: varies.

text: The

text: above

text: analysis

text: for

text: AWGN

text: channels

text: need

text: to

text: be

text: generalized.

text: Since

text: the

text: receiver

text: knows

text: the

text: CSI,

text: the

text: fading

text: effect

text: can

text: be

text: removed

text: by

text: multiplying

text: Y

text: with

text: We

text: deﬁne

text: the

text: fading

text: H

text: mod-Λ

text: channel

text: as

text: follows.

text: modfading.eps

text: Fig.

text: A

text: block

text: diagram

text: of

text: the

text: fading

text: mod-Λ

text: channel.

text: V

text: Deﬁnition

text: (fading

text: mod-Λ

text: channel):

text: A

text: fading

text: mod-Λ

text: channel

text: is

text: a

text: fading

text: channel

text: with

text: an

text: input

text: in

text: (Λ),

text: and

text: V

text: an

text: output

text: being

text: scaled

text: by

text: before

text: the

text: mod-

text: (Λ)

text: operation.

text: A

text: block

text: diagram

text: of

text: this

text: model

text: is

text: shown

text: in

text: Fig.

text: H

text: May

text: DRAFT

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text: 15 (cid:48) (cid:48) Since the Λ/Λ channel is symmetric, it is easy to check that the Λ/Λ fading channel is symmetric as well. | (cid:48)| (cid:48) Z Z Moreover, if Λ/Λ = 2, the Λ/Λ fading channel is a BMSC. Taking the binary partition /2 as an example, Z Z ∈ { } the input of the /2 fading channel is X 0, 1 , and a permutation φ over the outputs (y˜, h) is deﬁned such − Z | | that φ(y˜, h) = ([y˜ 1] mod 2 , h). Check that P | (y˜, h 0) = P | (φ(y˜, h) 1). Y˜,H X Y˜,H X It is now clear that polar lattices can be constructed to achieve the (ergodic) Poltyrev capacity of the i.i.d. fading channels, as we did for the AWGN channel in Sect. . Recall that the Poltyrev capacity C∞ of a general additive-noise channel is deﬁned as the capacity per unit volume in [25, Theorem 6.3.1]. For the independent AWGN channel, we have (cid:18) (cid:19) − 1 1 C∞ = h(σ2) = log , (34) 2 2πeσ2 where h(σ2) denotes the differential entropy of a Gaussian random variable with variance σ2. For independent fading channels, C∞ is ge(cid:20)ne(cid:18)raliz(cid:19)ed(cid:21)as [36](cid:20) (cid:18) (cid:19)(cid:21) − σ2 1 h2 C∞ = E h = E log . (35) h h2 h 2 2πeσ2 In the special case of Rayleigh fading, (cid:90) (cid:18) (cid:19) − (cid:90)h − h(cid:18)2 1 (cid:18) 2πeσ(cid:19)2 (cid:19) C∞ = σ2 e 2σh2 2 log h2 dh h h −1 − 2πeσ2 − = (cid:18)e t log (cid:19) log t dt (36) t=2hσ2h2 2 t 2σh2 −1 · eζ = log 2πeσ2 , (cid:82) 2 2σ2 h (cid:16) (cid:17) − ∞ − where ζ = e x ln xdx is the Euler-Mascheroni constant. 0 − · To approach the Poltyrev capacity 1 log 2πeσ2 eζ , we construct polar lattices according to the following 2 (cid:104) (cid:16)2σ2 (cid:17)(cid:105) h three design criteria: (cid:104) (cid:16) (cid:17)(cid:105) (a) The top lattice Λ gives negligible capacity E C Λ, σ2 . h h2 (cid:48) (cid:48) (b) The bottom lattice Λ has a small error probability E P Λ , σ2 . h e h2 (cid:16) (cid:17) C (c) Each component polar code is a capacity-approaching code for the Λ − /Λ fading channel. (cid:16) (cid:17) (cid:16) (cid:17) (cid:96) (cid:96) 1 (cid:96) ≈ For criterion (a), we pick a top lattice Λ for a large channel gain h such that h Λ, σ2 log V (Λ). By Remark l h2 ≥ ≤ ≤ l 5, h Λ, σ2 h Λ, σ2 for 0 h h . h2 h2 l l May 28, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Since

text: the

text: Λ/Λ

text: channel

text: is

text: symmetric,

text: it

text: is

text: easy

text: to

text: check

text: that

text: the

text: Λ/Λ

text: fading

text: channel

text: is

text: symmetric

text: as

text: well.

text: Z

text: Z

text: Moreover,

text: if

text: Λ/Λ

text: the

text: Λ/Λ

text: fading

text: channel

text: is

text: a

text: BMSC.

text: Taking

text: the

text: binary

text: partition

text: as

text: an

text: example,

text: Z

text: Z

text: the

text: input

text: of

text: the

text: fading

text: channel

text: is

text: X

text: and

text: a

text: permutation

text: φ

text: over

text: the

text: outputs

text: (y˜,

text: h)

text: is

text: deﬁned

text: such

text: Z

text: that

text: φ(y˜,

text: h)

text: ([y˜

text: mod

text: h).

text: Check

text: that

text: P

text: (y˜,

text: h

text: P

text: (φ(y˜,

text: h)

text: Y˜,H

text: X

text: Y˜,H

text: X

text: It

text: is

text: now

text: clear

text: that

text: polar

text: lattices

text: can

text: be

text: constructed

text: to

text: achieve

text: the

text: (ergodic)

text: Poltyrev

text: capacity

text: of

text: the

text: i.i.d.

text: fading

text: channels,

text: as

text: we

text: did

text: for

text: the

text: AWGN

text: channel

text: in

text: Sect.

text: Recall

text: that

text: the

text: Poltyrev

text: capacity

text: C∞

text: of

text: a

text: general

text: additive-noise

text: channel

text: is

text: deﬁned

text: as

text: the

text: capacity

text: per

text: unit

text: volume

text: in

text: Theorem

text: For

text: the

text: independent

text: AWGN

text: channel,

text: we

text: have

text: C∞

text: log

text: where

text: denotes

text: the

text: differential

text: entropy

text: of

text: a

text: Gaussian

text: random

text: variable

text: with

text: variance

text: For

text: independent

text: fading

text: channels,

text: C∞

text: is

text: σ2

text: C∞

text: E

text: h

text: E

text: log

text: h

text: h

text: In

text: the

text: special

text: case

text: of

text: Rayleigh

text: fading,

text: C∞

text: σ2

text: e

text: log

text: dh

text: h

text: h

text: −1

text: t

text: log

text: log

text: t

text: dt

text: t

text: −1

text: eζ

text: log

text: h

text: where

text: ζ

text: e

text: x

text: ln

text: xdx

text: is

text: the

text: Euler-Mascheroni

text: constant.

text: To

text: approach

text: the

text: Poltyrev

text: capacity

text: log

text: eζ

text: we

text: construct

text: polar

text: lattices

text: according

text: to

text: the

text: following

text: h

text: three

text: design

text: criteria:

text: (a)

text: The

text: top

text: lattice

text: Λ

text: gives

text: negligible

text: capacity

text: E

text: C

text: Λ,

text: σ2

text: h

text: (b)

text: The

text: bottom

text: lattice

text: Λ

text: has

text: a

text: small

text: error

text: probability

text: E

text: P

text: Λ

text: σ2

text: h

text: e

text: C

text: (c)

text: Each

text: component

text: polar

text: code

text: is

text: a

text: capacity-approaching

text: code

text: for

text: the

text: Λ

text: /Λ

text: fading

text: channel.

text: For

text: criterion

text: (a),

text: we

text: pick

text: a

text: top

text: lattice

text: Λ

text: for

text: a

text: large

text: channel

text: gain

text: h

text: such

text: that

text: h

text: Λ,

text: σ2

text: log

text: V

text: (Λ).

text: By

text: Remark

text: l

text: l

text: h

text: Λ,

text: σ2

text: h

text: Λ,

text: σ2

text: for

text: h

text: h

text: l

text: l

text: May

text: DRAFT

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text: ##LTLine##

text: ##LTLine##


text: 8 · where h ( ) is the binary entropy function. 2 The fading channel W˜ is then quantized according to C[LR(y, h)]. Let µ = 2Q be the alphabet size of the { ≥ ≥ } degraded channel output alphabet. The set y 0, h 0 is divided into Q subsets (cid:26) (cid:27) − ≥ ≥ i 1 ≤ i A = y 0, h 0 : C[LR(y, h)] < , (19) i Q Q ≤ ≤ for 1 i Q. Typical boundaries of A are depicted in Fig. 2. The outputs in A are mapped to one symbol, i i and W˜ is quantized to a mixture of Q BSC(cid:82)s with the crossover probability (cid:82) (cid:82) | − P | (y, h 1)dydh p = |Ai Y,H X | − . (20) i P | (y, h + 1)dydh + P | (y, h 1)dydh Ai Y,H X Ai Y,H X Note that pi can(cid:40)be numerically evalu(cid:18)ated. Since LR(y, h(cid:19)) = e2σy2h , Ai is (cid:18)rewritten as (cid:19)(cid:41) ≥ ≥ σ2 1 − ≤ σ2 1 − Ai = y 0, h 0 : 2 ln h−1(Q−i+1) 1 yh < 2 ln h−1(Q−i) 1 . (21) 2 Q 2 Q AiBoundary.eps Fig. 2. Typical boundaries of Ai for channel quantization. Let δ and δ denote the two bounds in (21). We have 1 2 (cid:90) (cid:90) (cid:90) (cid:82) Ai PY,H|X(y, h| + 1)dydh = 0∞ σhh2 e−2hσ2h2 dh δh1δh2 √21πσ2 e−(y2−σh2)2 dy, (22) | − and P | (y, h 1)dydh is calculated similarly. Ai Y,H X May 28, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: h

text: is

text: the

text: binary

text: entropy

text: function.

text: The

text: fading

text: channel

text: W˜

text: is

text: then

text: quantized

text: according

text: to

text: C[LR(y,

text: h)].

text: Let

text: µ

text: be

text: the

text: alphabet

text: size

text: of

text: the

text: degraded

text: channel

text: output

text: alphabet.

text: The

text: set

text: y

text: h

text: is

text: divided

text: into

text: Q

text: subsets

text: i

text: i

text: A

text: y

text: h

text: C[LR(y,

text: h)]

text: i

text: Q

text: Q

text: for

text: i

text: Q.

text: Typical

text: boundaries

text: of

text: A

text: are

text: depicted

text: in

text: Fig.

text: The

text: outputs

text: in

text: A

text: are

text: mapped

text: to

text: one

text: symbol,

text: i

text: i

text: and

text: W˜

text: is

text: quantized

text: to

text: a

text: mixture

text: of

text: Q

text: with

text: the

text: crossover

text: probability

text: P

text: (y,

text: h

text: p

text: |Ai

text: Y,H

text: X

text: i

text: P

text: (y,

text: h

text: P

text: (y,

text: h

text: Ai

text: Y,H

text: X

text: Ai

text: Y,H

text: X

text: Note

text: that

text: pi

text: numerically

text: Since

text: LR(y,

text: Ai

text: is

text: as

text: σ2

text: σ2

text: Ai

text: y

text: h

text: ln

text: yh

text: ln

text: Q

text: Q

text: AiBoundary.eps

text: Fig.

text: Typical

text: boundaries

text: of

text: Ai

text: for

text: channel

text: quantization.

text: Let

text: δ

text: and

text: δ

text: denote

text: the

text: two

text: bounds

text: in

text: We

text: have

text: Ai

text: PY,H|X(y,

text: h|

text: 0∞

text: dh

text: √21πσ2

text: dy,

text: and

text: P

text: (y,

text: h

text: is

text: calculated

text: similarly.

text: Ai

text: Y,H

text: X

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 Let W˜ denote the quantized channel from W˜ after the degrading transform. By [2, Lemma 13], the difference Q between the two channel capacities is upper-bounded by 1 . A comparison between C(W˜ ) and C(W˜ ) for different Q Q SNR when Q = 128 is shown in Fig. 3. When Q is sufﬁciently large, we can use W˜ to approximate W˜ in the Q construction of polar codes. The size of the output alphabet after the degrading merging is no more than 2Q. CDiff_WV.eps → Fig. 3. Comparisons between C(W˜Q) and C(W˜ ), and between C(V˜Q) and C(V˜), when Q = 128. Here W˜ denotes the channel X (Y,H), → and V˜ denotes the channel X Y , i.e., the two channel models when the receiver knows the CSI and the CDI, respectively. W˜Q and V˜Q denote the quantized version of W˜ and V˜, respectively. The proof of the following theorem can be adapted from [2]. We omit it for brevity. → Theorem 1: Let W˜ : X (Y, H) be a binary-input i.i.d. fading channel. Let N denote the block length and µ = 2Q denote the limit of the size of output alphabet. A polar code constructed by the degrading merging algorithm achieves the capacity C(W˜ ) when N and µ are both sufﬁciently large. The block error probability under − SC decoding is upper-bounded by N2 Nβ for 0 < β < 1. 2 Simulation result of polar codes with different block length for the binary-input Rayleigh fading channel with CSI available to the receiver are shown in Fig. 4, where SNR = 5 dB and C(W˜ ) = 0.6709. The performance can be further improved by using more sophisticated decoding algorithms [37], [38]. Remark 3. It has been pointed out in [17] that polar codes for the Rayleigh fading channel with known CDI suffer a penalty for not having complete information. The statement can be seen clearly from our construction. Treating → → H as part of channel outputs, the binary channel X Y is degraded with respect to the channel X (Y, H), ≥ → and I(X; Y, H) I(X; Y ). Let V˜ denote the channel X Y . The channel transition PDF of V˜ is written as (cid:90) | | P | (y x) = P | (y, h x)dh, (23) Y X Y,H X h May 28, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Let

text: W˜

text: denote

text: the

text: quantized

text: channel

text: from

text: W˜

text: after

text: the

text: degrading

text: transform.

text: By

text: Lemma

text: the

text: difference

text: Q

text: between

text: the

text: two

text: channel

text: capacities

text: is

text: upper-bounded

text: by

text: A

text: comparison

text: between

text: C(W˜

text: and

text: C(W˜

text: for

text: different

text: Q

text: Q

text: SNR

text: when

text: Q

text: is

text: shown

text: in

text: Fig.

text: When

text: Q

text: is

text: sufﬁciently

text: large,

text: we

text: can

text: use

text: W˜

text: to

text: approximate

text: W˜

text: in

text: the

text: Q

text: construction

text: of

text: polar

text: codes.

text: The

text: size

text: of

text: the

text: output

text: alphabet

text: after

text: the

text: degrading

text: merging

text: is

text: no

text: more

text: than

font: QOQVQG+NimbusMonL-Regu

text: Fig.

text: Comparisons

text: between

text: C(W˜Q)

text: and

text: C(W˜

text: and

text: between

text: C(V˜Q)

text: and

text: C(V˜),

text: when

text: Q

text: Here

text: W˜

text: denotes

text: the

text: channel

text: X

text: (Y,H),

text: and

text: V˜

text: denotes

text: the

text: channel

text: X

text: Y

text: i.e.,

text: the

text: two

text: channel

text: models

text: when

text: the

text: receiver

text: knows

text: the

text: CSI

text: and

text: the

text: CDI,

text: respectively.

text: W˜Q

text: and

text: V˜Q

text: denote

text: the

text: quantized

text: version

text: of

text: W˜

text: and

text: V˜,

text: respectively.

text: The

text: proof

text: of

text: the

text: following

text: theorem

text: can

text: be

text: adapted

text: from

text: We

text: omit

text: it

text: for

text: brevity.

text: Theorem

text: Let

text: W˜

text: X

text: (Y,

text: H)

text: be

text: a

text: binary-input

text: i.i.d.

text: fading

text: channel.

text: Let

text: N

text: denote

text: the

text: block

text: length

text: and

text: µ

text: denote

text: the

text: limit

text: of

text: the

text: size

text: of

text: output

text: alphabet.

text: A

text: polar

text: code

text: constructed

text: by

text: the

text: degrading

text: merging

text: algorithm

text: achieves

text: the

text: capacity

text: C(W˜

text: when

text: N

text: and

text: µ

text: are

text: both

text: sufﬁciently

text: large.

text: The

text: block

text: error

text: probability

text: under

text: SC

text: decoding

text: is

text: upper-bounded

text: by

text: Nβ

text: for

text: β

text: Simulation

text: result

text: of

text: polar

text: codes

text: with

text: different

text: block

text: length

text: for

text: the

text: binary-input

text: Rayleigh

text: fading

text: channel

text: with

text: CSI

text: available

text: to

text: the

text: receiver

text: are

text: shown

text: in

text: Fig.

text: where

text: SNR

text: dB

text: and

text: C(W˜

text: The

text: performance

text: can

text: be

text: further

text: improved

text: by

text: using

text: more

text: sophisticated

text: decoding

text: algorithms

text: Remark

text: It

text: has

text: been

text: pointed

text: out

text: in

text: that

text: polar

text: codes

text: for

text: the

text: Rayleigh

text: fading

text: channel

text: with

text: known

text: CDI

text: suffer

text: a

text: penalty

text: for

text: not

text: having

text: complete

text: information.

text: The

text: statement

text: can

text: be

text: seen

text: clearly

text: from

text: our

text: construction.

text: Treating

text: H

text: as

text: part

text: of

text: channel

text: outputs,

text: the

text: binary

text: channel

text: X

text: Y

text: is

text: degraded

text: with

text: respect

text: to

text: the

text: channel

text: X

text: (Y,

text: H),

text: and

text: I(X;

text: Y,

text: H)

text: I(X;

text: Y

text: Let

text: V˜

text: denote

text: the

text: channel

text: X

text: Y

text: The

text: channel

text: transition

text: PDF

text: of

text: V˜

text: is

text: written

text: as

text: P

text: (y

text: x)

text: P

text: (y,

text: h

text: x)dh,

text: Y

text: X

text: Y,H

text: X

text: h

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 JOHN A. BALDWIN AND STEVEN SIVEK of this complex. { } Suppose now that (M, Γ, ξ) is a sutured contact manifold and that (S, P, h, c , . . . , c ) is a 1 n compatible partial open book. Let Σ be the surface formed by attaching 1-handles H , . . . , H 1 n to S, where the feet of H are attached along the endpoints of c . Orient Σ so that the induced i i orientation on S as a subsurface of Σ is opposite the given orientation on S. For i = 1, . . . , n, let α and β be embedded curves in Σ such that: i i • α is the union of c with a core of H , and • i i i β is the union of h(c ) with a core of H . i i i ⊂ We require that these curves intersect in the region H Σ in the manner shown in Figure i − − 2. Then (Σ, β, α) is an admissible sutured Heegaard diagram for ( M, Γ), as shown in [19]. For each i = 1, . . . , n, let ci be the unique intersection point between α and β in H , and i i i deﬁne { } ∈ T ∩ T ⊂ (4) c = c1, . . . , cn Symn(Σ). β α In [19], the contact invariant c (ξ) is given by HF ∈ − − c (ξ) := [c] SF H( M, Γ). HF − βi αi S handlecurves.pdf ci H i Figure 2. The handle H , the curves α and β , and the intersection point ci. i i i Finally, we recall the deﬁnition of Heegaard Floer homology with local coeﬃcients and we review the construction of Ozsv´ath and Szab´o’s contact invariant for closed manifolds therein. Suppose Y is a closed, oriented 3-manifold, and η is an oriented, embedded curve in Y . To deﬁne the Heegaard Floer homology of Y with local coeﬃcient system associated with η, one starts with an admissible pointed Heegaard diagram { } { } (Σ, α = α , . . . , α , β = β , . . . , β , z) 1 n 1 n for Y . We may view η as a (possibly non-embedded) curve on Σ. The chain complex CF +(α, β; Γ )(cid:18):= CF +(Σ,(cid:19)α, β, z; Γ ) (cid:77) η η is the Λ[U]-module − Λ[U, U 1] (cid:104) (cid:105) · x , ∈ ∩ U Λ[U] x Tα Tβ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JOHN

text: A.

text: BALDWIN

text: AND

text: STEVEN

text: SIVEK

text: of

text: this

text: complex.

text: Suppose

text: now

text: that

text: (M,

text: Γ,

text: ξ)

text: is

text: a

text: sutured

text: contact

text: manifold

text: and

text: that

text: (S,

text: P,

text: h,

text: c

text: c

text: is

text: a

text: n

text: compatible

text: partial

text: open

text: book.

text: Let

text: Σ

text: be

text: the

text: surface

text: formed

text: by

text: attaching

text: H

text: H

text: n

text: to

text: S,

text: where

text: the

text: feet

text: of

text: H

text: are

text: attached

text: along

text: the

text: endpoints

text: of

text: c

text: Orient

text: Σ

text: so

text: that

text: the

text: induced

text: i

text: i

text: orientation

text: on

text: S

text: as

text: a

text: subsurface

text: of

text: Σ

text: is

text: opposite

text: the

text: given

text: orientation

text: on

text: S.

text: For

text: i

text: n,

text: let

text: α

text: and

text: β

text: be

text: embedded

text: curves

text: in

text: Σ

text: such

text: that:

text: i

text: i

text: α

text: is

text: the

text: union

text: of

text: c

text: with

text: a

text: core

text: of

text: H

text: and

text: i

text: i

text: i

text: β

text: is

text: the

text: union

text: of

text: h(c

text: with

text: a

text: core

text: of

text: H

text: i

text: i

text: i

text: We

text: require

text: that

text: these

text: curves

text: intersect

text: in

text: the

text: region

text: H

text: Σ

text: in

text: the

text: manner

text: shown

text: in

text: Figure

text: i

text: Then

text: (Σ,

text: β,

text: α)

text: is

text: an

text: admissible

text: sutured

text: Heegaard

text: diagram

text: for

text: M,

text: Γ),

text: as

text: shown

text: in

text: For

text: each

text: i

text: n,

text: let

text: ci

text: be

text: the

text: unique

text: intersection

text: point

text: between

text: α

text: and

text: β

text: in

text: H

text: and

text: i

text: i

text: i

text: deﬁne

text: T

text: T

text: c

text: cn

text: Symn(Σ).

text: β

text: α

text: In

text: the

text: contact

text: invariant

text: c

text: (ξ)

text: is

text: given

text: by

text: HF

text: c

text: (ξ)

text: [c]

text: SF

text: H(

text: M,

text: Γ).

text: HF

text: βi

text: αi

text: S

text: handlecurves.pdf

text: ci

text: H

text: i

text: Figure

text: The

text: handle

text: H

text: the

text: curves

text: α

text: and

text: β

text: and

text: the

text: intersection

text: point

text: ci.

text: i

text: i

text: i

text: Finally,

text: we

text: recall

text: the

text: deﬁnition

text: of

text: Heegaard

text: Floer

text: homology

text: with

text: local

text: coeﬃcients

text: and

text: we

text: review

text: the

text: construction

text: of

text: Ozsv´ath

text: and

text: Szab´o’s

text: contact

text: invariant

text: for

text: closed

text: manifolds

text: therein.

text: Suppose

text: Y

text: is

text: a

text: closed,

text: oriented

text: and

text: η

text: is

text: an

text: oriented,

text: embedded

text: curve

text: in

text: Y

text: To

text: deﬁne

text: the

text: Heegaard

text: Floer

text: homology

text: of

text: Y

text: with

text: local

text: coeﬃcient

text: system

text: associated

text: with

text: η,

text: one

text: starts

text: with

text: an

text: admissible

text: pointed

text: Heegaard

text: diagram

text: (Σ,

text: α

text: α

text: α

text: β

text: β

text: β

text: z)

text: n

text: n

text: for

text: Y

text: We

text: may

text: view

text: η

text: as

text: a

text: (possibly

text: non-embedded)

text: curve

text: on

text: Σ.

text: The

text: chain

text: complex

text: CF

text: +(α,

text: β;

text: Γ

text: CF

text: β,

text: z;

text: Γ

text: η

text: η

text: is

text: the

text: Λ[U]-module

text: Λ[U,

text: U

text: x

text: U

text: Λ[U]

text: x

text: Tα

text: Tβ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 JOHN A. BALDWIN AND STEVEN SIVEK denote the intersection point in top Maslov grading. Then the map − − → − (cid:48) HF +( W ; Γ ) : HF +( Y ; Γ ) HF +( Y ; Γ ) ν η η is induced by the chain map → f+ : CF +(β, α; Γ ) CF +(γ, α; Γ ), γ,β,α;Γν(cid:88) (cid:88) η η deﬁned on [x, i] by M · − · · f+ ([x, i]) = # (φ) [y, i n (φ)] t∂α(φ) η, γ,β,α;Γν ∈T ∩T ∈ z y γ α φ π2(Θ,x,y) µ(φ)=0 where π (Θ, x, y) is the set of homotopy classes of Whitney triangles with vertices at Θ, x, y, M 2 and (φ) is the moduli space of holomorphic representatives of φ. This map is an invariant ∈ − − Z of the class [ν] H ( W, ∂W ; ). 2 3. Proof of Main Theorem Suppose (M, Γ, ξ) is a sutured contact manifold. The goal of this section is to prove Theorem 1.12 of the introduction, a more precise version of which is stated below. The Main Theorem of this paper, Theorem 1.2, then follows immediately, as explained in the introduction. Theorem 1.12. There exists a contact closure (Y, R, ξ¯) of (M, Γ, ξ) with an oriented curve ⊂ η R which is dual to an essential curve in the associated auxiliary surface T , for which there is an isomorphism of Λ-modules − − ⊗ → − |− F : SF H( M, Γ) Λ HF +( Y R; Γ ), η ⊗ sending c (ξ) 1 to c (ξ¯). HF HF We prove Theorem 1.12 according to the strategy outlined in the introduction. In particular, as we show in Subsection , Theorem 1.12 follows from Theorems 3.4 and 3.5. Fix a partial open book { } (S, P, h, c , . . . , c ) 1 n compatible with (M, Γ, ξ). Recall that an important ingredient in the proof of Theorem 1.12 − involves understanding the map (denoted in the introduction by G) induced by W , where W is the 2-handle cobordism from a closure of H(S) to a closure of (M, Γ), corresponding to surgery on the ∂H(S)-framed curves × { } ∪ × − ∪ × {− } ⊂ × − (5) s = (c 1 ) (∂c [ 1, 1]) (h(c ) 1 ) ∂(S [ 1, 1]) i i i i in the closure of H(S). In order to eventually understand this map (see Theorem 3.4), we − ﬁrst describe a pointed Heegaard triple diagram for the cobordism W .

text: JOHN

text: A.

text: BALDWIN

text: AND

text: STEVEN

text: SIVEK

text: denote

text: the

text: intersection

text: point

text: in

text: top

text: Maslov

text: grading.

text: Then

text: the

text: map

text: HF

text: W

text: Γ

text: HF

text: Y

text: Γ

text: HF

text: Y

text: Γ

text: ν

text: η

text: η

text: is

text: induced

text: by

text: the

text: chain

text: map

text: f+

text: CF

text: +(β,

text: α;

text: Γ

text: CF

text: +(γ,

text: α;

text: Γ

text: η

text: η

text: deﬁned

text: on

text: [x,

text: i]

text: by

text: M

text: f+

text: ([x,

text: i])

text: (φ)

text: [y,

text: i

text: n

text: (φ)]

text: t∂α(φ)

text: η,

text: γ,β,α;Γν

text: ∈T

text: ∩T

text: z

text: y

text: γ

text: α

text: φ

text: where

text: π

text: (Θ,

text: x,

text: y)

text: is

text: the

text: set

text: of

text: homotopy

text: classes

text: of

text: Whitney

text: triangles

text: with

text: vertices

text: at

text: Θ,

text: x,

text: y,

text: M

text: and

text: (φ)

text: is

text: the

text: moduli

text: space

text: of

text: holomorphic

text: representatives

text: of

text: φ.

text: This

text: map

text: is

text: an

text: invariant

text: Z

text: of

text: the

text: class

text: [ν]

text: H

text: W,

text: ∂W

text: Proof

text: of

text: Main

text: Theorem

text: Suppose

text: (M,

text: Γ,

text: ξ)

text: is

text: a

text: sutured

text: contact

text: manifold.

text: The

text: goal

text: of

text: this

text: section

text: is

text: to

text: prove

text: Theorem

text: of

text: the

text: introduction,

text: a

text: more

text: precise

text: version

text: of

text: which

text: is

text: stated

text: below.

text: The

text: Main

text: Theorem

text: of

text: this

text: paper,

text: Theorem

text: then

text: follows

text: immediately,

text: as

text: explained

text: in

text: the

text: introduction.

text: Theorem

text: There

text: exists

text: a

text: contact

text: closure

text: (Y,

text: R,

text: ξ¯)

text: of

text: (M,

text: Γ,

text: ξ)

text: with

text: an

text: oriented

text: curve

text: η

text: R

text: which

text: is

text: dual

text: to

text: an

text: essential

text: curve

text: in

text: the

text: associated

text: auxiliary

text: surface

text: T

text: for

text: which

text: there

text: is

text: an

text: isomorphism

text: of

text: Λ-modules

text: F

text: SF

text: H(

text: M,

text: Γ)

text: Λ

text: HF

text: Y

text: R;

text: Γ

text: η

text: sending

text: c

text: (ξ)

text: to

text: c

text: (ξ¯).

text: HF

text: HF

text: We

text: prove

text: Theorem

text: according

text: to

text: the

text: strategy

text: outlined

text: in

text: the

text: introduction.

text: In

text: particular,

text: as

text: we

text: show

text: in

text: Subsection

text: Theorem

text: follows

text: from

text: Theorems

text: and

text: Fix

text: a

text: partial

text: open

text: book

text: (S,

text: P,

text: h,

text: c

text: c

text: n

text: compatible

text: with

text: (M,

text: Γ,

text: ξ).

text: Recall

text: that

text: an

text: important

text: ingredient

text: in

text: the

text: proof

text: of

text: Theorem

text: involves

text: understanding

text: the

text: map

text: (denoted

text: in

text: the

text: introduction

text: by

text: G)

text: induced

text: by

text: W

text: where

text: W

text: is

text: the

text: cobordism

text: from

text: a

text: closure

text: of

text: H(S)

text: to

text: a

text: closure

text: of

text: (M,

text: Γ),

text: corresponding

text: to

text: surgery

text: on

text: the

text: ∂H(S)-framed

text: curves

text: s

text: (c

text: (∂c

text: (h(c

text: ∂(S

text: i

text: i

text: i

text: i

text: in

text: the

text: closure

text: of

text: H(S).

text: In

text: order

text: to

text: eventually

text: understand

text: this

text: map

text: (see

text: Theorem

text: we

text: ﬁrst

text: describe

text: a

text: pointed

text: Heegaard

text: triple

text: diagram

text: for

text: the

text: cobordism

text: W