# Weight enumerator of the Hamming code

Let $H_r$ be the usual Hamming code of length $2^r-1$. What is the weight enumerator of $H_r^\perp$? Using this find an expression for the weight enumerator of $H_r$. (we are in binary case)

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## 1 Answer

Hints:

1. The duals of the Hamming codes are the Simplex codes. The Simplex codes are "constant weight codes", since all nonzero codewords have the same weight. Can you find the weight enumerator of the Simplex codes from this information?

2. The weight enumerators of the primal and the dual code are related by the MacWilliams identities. Use this to translate the weight enumerator of the Simplex codes into the weight enumerator of the Hamming codes.

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thank you azimut i will try to use your hints – juliet May 10 '13 at 21:16