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

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