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[Solved]: Existence of Hamming code

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We are given a number $n \geq 3$ and we know that the Hamming bound is satisfied. Does this imply that there is a Hamming code with length $\frac{q^r-1}{q-1}$, dimension $\frac{q^r-1}{q-1}-r$ and Hamming-distance $3$ or do we have to construct such a code, given that the bound is satisfied?

Asked By : Evinda

Answered By : Yuval Filmus

The Hamming bound is an upper bound on the size of codes. It's not a tight bound in general, though in some specific cases it is achievable. Codes achieving the Hamming bound are called perfect codes.

When the alphabet size is a prime power, all perfect codes are known: apart from some trivial cases (repetition codes, codes with one codeword, and codes containing all possible codewords), they share the same parameters as a Hamming code or a Golay code.

What this means is that if your alphabet size is a prime power and you have parameters satisfying the Hamming bound, then a code with these parameters actually exists only in the cases listed above. In any other case, the bound isn't tight.

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