TITLE: Reed-Muller-Codes
QUESTION [5 upvotes]: Let $F$ be the field with two elements, $V_m=F^{2^m}$.Let $R(r, m)\subset V_m$ be the 
binary Reed-Muller Code. Define $R_m:=R(1, m)$. Then the dimension of $R_m$ is 
$1+m$ and its minimal distance is $d(R_m)=2^{m-1}$. (Cf. for example the book of Luetkebohmert). Hence the information rate is $I(R_m)=\frac{1+m}{2^m}$ while the relative
minimal distance of $R_m$ is $rd(R_m)=\frac{1}{2}$.
Now $R_5$ has the same relative minimal distance as $R_4$ (and thus should have approximately the same error correction abilities), while the information rate of $R_4$ is much better ($I(R_4)\approx 0.31$ while $I(R_5)\approx 0.19$).
I furthermore read in Luetkebohmert (and elsewhere) that $R_5$ was used in practice, e.g. during a Mariner space shuttle mission.
Question: Given the values above, why would anybody use $R_5$ and not $R_4$?
(I hope this question is not too easy for MO, but I do not see the answer at the moment.)
Edit: It seems that the code $R_5$ might have been used for Mariner, because then one has really one code word per pixel of the image. (Compare the comments)
Still let me note that the $R_m$ all have the same relative minimal distance, that there
exists an efficient (poly time?) algorithm to decode them, and that 
$I(R_m)$ tends rapidly to zero when $n\to\infty$. And I find this a bit strange: For an "interesting" or even "celebrated" family of codes $(C_m)_m$, which all have the same relative minimal distance, I would naively have expected something like $(I(C_m))_m$ should be a decreasing (but bounded or even converging) sequence.

REPLY [6 votes]: This began as a comment on the comments by @Chris and @Jyrki but posted as an answer because I don't have sufficient reputation on this site.
The standard decoding algorithm for Reed-Muller codes uses majority-logic
decoding which in turn requires hard-decision demodulation. That is, the
input to the decoder is a $32$-bit vector that is obtained from a demodulator
a device that  processes $\mathbf x$, a vector of $32$ real numbers,
to produce a $32$-bit sequence.
However, for first-order 
Reed-Muller codes, it is possible to use soft-decision demodulation and decoding in
Euclidean space, that is, the decoder finds the (binary) codeword that is closest
in the Euclidean metric to $\mathbf x$.  This is because the $64$ codewords of $R_5$, regarded as vectors in  $\{+1, -1\}^n$ (instead of $\{0, 1\}^n$) are just the rows of a $32\times 32$ Hadamard matrix $H$ (or 
the negatives of the rows).  All that needs to be done is to compute 
$\mathbf y = \mathbf x H$.
The location of the entry with the largest magnitude in $\mathbf y$ together with
the sign of the entry, tells us which of the $64$
codewords is the nearest in Euclidean distance to $\mathbf x$.
If the codewords are arranged so that $H$ is in Sylvester form, then a fast
Hadamard transform algorithm (very similar to a Fast Fourier Transform algorithm)
can be used to compute $\mathbf x H$ using $160 = 32\log_2 32$ additions ans
subtractions.  In fact, this form of the computation was used in the Mariner 
program; the machine devised to carry out the calculations was referred to as 
the Green Machine.
An article describing many of the details of the coding for the Mariner
missions can be found in a conference proceedings published by Wiley
under the title Error-Correcting Codes, H. B. Mann (ed.), 1969. 
Some details, and a diagram of part of the Green Machine can be found
in Chapter 14 of MacWilliams and Sloane's The Theory of Error-Correcting
Codes, North-Holland, 1978.  I am sure that Jyrki has the latter on
his bookshelf.