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Consider binary words in {0,1}^n whose Hamming weight is at most some constant d. We want to select some of these words such that they form a group under addition. How many words can we choose at most?

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Your structure is known as a group anticode with maximal distance $d$. The obvious construction is to take all vectors of the form $(x_1,\ldots,x_d,0,\ldots,0)$, obtaining a group of size $2^d$. Ahlswede, Another diametric theorem in Hamming spaces: optimal group anticodes showed that this is optimal, and furthermore, the solution I described is unique up to isomorphism, except when $d = 2$, in which case another solution is $(0,0,0,\bar{0}),(1,1,0,\bar{0}),(1,0,1,\bar{0}),(0,1,1,\bar{0})$.

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  • $\begingroup$ Great help. Thanks. $\endgroup$ Apr 13 at 13:40

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