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It is not too difficult to find a binary code consisting of $8$ words, each $13$ bits long, keeping the distance between every pair of words at least $7$. I know it is not possible to find $9$ words with that property, but I could not prove it. Can anybody please help me?

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    Can you prove that the 8 word code is unique up to equivalence, i.e. linear? Then the claim would follow from Griesmer bound, because 3 is obviously a maximum dimension of a code of length 13 and minimum distance 7 (and the existence of a 9 word code would actually imply the existence of a 16 word code). What does the linear programming bound (=McEliece-Rodemich-Rumsey-Welch bound) tell in this case?2012-08-07

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