I need to choose $n$ samples of $x$-bit sequences. What is the criterion which will maximize the "information content" or ensure that the $n$ chosen bit sequences have maximal separation from each other (i.e. maximally distinct sequences)?
As an example, let's say we have 4-bits (i.e. $x = 4$). Therefore, the total number of bit sequences possible are 16, i.e. $ 0000, 0001, 0010, 0011, \dots , 1110, 1111$.
How can I choose $n$ samples that are somehow maximally distinct from each other? i.e. I'd like to choose $n$ bit patterns ($n \le 16$), e.g. $n=5$, that somehow 'span' as much of the space as possible, whilst being maximally spatially separated from each other (somehow the analogy of sobol/lhs/orthogonal sampling idea pops in my mind, but its all very nebulous at the moment, and not sure if this is the right idea indeed)?
Updated Question:
Based on Karl's comment below, let me refine the question a bit more. I'd like to choose $n$ bit patterns from the possible combinations such that they are all maximally separated from each other i.e. somehow, these $n$ samples must together aim to be as "spread out as possible" on the entire possible domain (i.e. all possible bit sequences)?
Your question is somewhat vague. Let me rephrase it slightly: from the set of $2^n$ binary tuples of length $n$, we want to select $M$ tuples such that the minimum Hamming distance among all the pairs is maximized.
This is a well known problem in the field of (binary) error correcting codes, and lot of work has been invested into in since 1950, mostly restricted to linear codes (i.e. those where the $M$ tuples -"codewords"- form a vector space - in this case $M=2^k$ for some $k<n$). There are no simple general answers, but there are lots of bounds, and some recipes. See eg the BCH codes.
For small values of $n$ (and not necessarily linear codes), there are exhaustive tables, such as this one.