On a recursive structure of binary Hamming codes

We show that binary Hamming codes can be constructed recursively from binary block repetition codes with a parity bit. The recursive structure can be used to obtain the input-output weight enumerator for any block-length whereas brute-force enumeration can only be used for short block-length. The input-output weight enumerator of binary Hamming codes can be also derived using the split-weight enumerator of dual codes and the generalized MacWilliams identity. Hence, the bit-error rate of binary Hamming codes with binary antipodal signaling and hard-decision demodulation used on additive white Gaussian noise channels can be evaluated exactly. Numerically computed coding gain of Hamming codes reveals a surprising fact that the coding gain is minimum for a particular small value of SNR and the minimum coding gain is always negative. This contradicts statements found in the literature. Finally, we obtain the input-output weight enumerator of extended binary Hamming codes and we show that these codes also meet the Hamming bound although they are not perfect.