TITLE: find a group of lowest N numbers so that no 2 pairs have the same bitwise or
QUESTION [5 upvotes]: I am trying to find the lowest group of N numbers (i.e. N=1000)
so that no 2 pairs from the group have the same bit-wise or.
more specific need to find a group
$A = \{a_1,a_2,a_3,..,a_N\} $
such that  for any $ 1\le i,j,k,m \le n $ 
$ i \ne j , k \ne m $
$OR(a_i,a_j) \ne OR(a_k,a_m)$
$a_1 .. a_n$ lowest as possable
example for N=3 
A = {0,1,2}
as Or(0,1) = 1 , Or(0,2) = 2 , Or(1,2) = 3

REPLY [5 votes]: The question "Can we find $1000$ numbers in the range $[0,2^{n}-1]$?" can be reworded as "Can we find $1000$ subsets of $\{1,\dots, n\}$ such that 

There do not exist distinct $A,B,C,D$ with $A \cup B=C \cup D$
There do not exist distinct $A,B,C$ with $A \cup B=A \cup C$

In extremal set theory, such a collection of subsets is called a strongly union free family.  Coppersmith and Shearer have some asymptotic constructions of fairly large (roughly $2^{0.31n}$ subsets) collections satisfying this property.  I'm not too familiar with the details of the constructions, but it might be possible to extract something in the non-asymptotic regime from them.