What is the highest achievable pair-wise Hamming distance $d$ between all possible pairs from $k$ bitvectors each having a length of $n$ bits? The content of each bitvector can be arbitrary, only the distance has to be maximized (thus the ordering of the vectors is irrelevant).
One common application for this is in coding theory where you want to spread the code words in such a way to maximize error detection capability.
This is how far I got:
$k=2$ : 2nd vector = negated 1st vector
-> $d = n$
$k<=ld(n^2)+2$; n = power of two: (itteratively)
1st vector = 0....0
for i = 2 to k:
if i is even: vector i = negated vector i-1
if i is odd: vector where half the bits are set in packs of 2/(i+2) bits
e.g. $n=4$; $k=6$: 0000 1111 0011 1100 0101 1010
e.g. $n=8$; $k=8$: 00000000 11111111 00001111 11110000 00110011 11001100 010101010 10101010
-> $d = n/2$
I found an question with the same intent saying this is still an open question :(
Generating a binary code with maximized Hamming distance
Nonetheless the linked tabel from the posting gives a solution for one of my cases ($n=28$, $k= 16$): $d=8$