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{||-| Assume we associate a bitstring to every piece-square combination. That is what's usually done in chess programs; some codes are added for the side to move, castling rights, e.p. squares, etc. We obtain the code of a position by xor-ing the codes of all the pieces contained in it.
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The minimum and average [[Population Count#HammingDistance|Hamming Distance]] over all Zobrist keys was often considered as "quality"-measure of the keys. However, maximizing the minimal hamming distance leads to very poor Zobrist keys. As long the minimum hamming distance is greater zero, [https://en.wikipedia.org/wiki/Linear_independence linear independence] (that is a small subset of all keys doesn't xor to zero), is much more important than hamming distance as explained by [[Sven Reichard]] <ref>[https://www.stmintz.com/ccc/index.php?id=200622 Re: About random numbers and hashing] by [[Sven Reichard]], [[CCC]], December 05, 2001</ref> :
What we want to avoid is collisions at nodes close to the root. For nodes close to the leaves the cost of recomputing the score is smaller. Hence we want to avoid that:
Summarizing I can say that I see no connection between the quality of hash codes and their Hamming distance. Using a good RNG like the one provided in GNU's stdlib will yield good hash codes ( you can actually prove that), and so I will take the codes as they are supplied by rand() or random() without messing with them and thereby most likely make them worse.
=See also=
