A coset weight count that proves that the simplex codes are not optimal for error correction

摘要

The number of cosets of weight 2/sup k-2/ or less are determined for the [2/sup k/-1, k, 2/sup k-1/] simplex code and a [2/sup k/-1, k, 2/sup k-1/-1] code obtained by a simple modification of the simplex code. The result proves that the [2/sup k/-1, k] simplex codes are not optimal for error correction on the binary symmetric channel with small bit error probability, p, (for k/spl ges/3). A proof that the modified code is better for all p, 0>p>1/2, is sketched.