On the size of optimal binary codes of length 9 and covering radius 1

The minimum number of codewords in a binary code with length n and covering radius R is denoted by K(n, R). The values of K (n, 1) are known up to len gth 8, and the corresponding optimal codes have been classified. It is know n that 57 less than or equal to K(9, 1) less than or equal to 62. In the cu rrent work, the lower bound is improved to settle K(9, 1) = 62. In the appr oach, which is computer-aided, possible distributions of codewords in subsp aces are refined until each subspace is of dimension zero (consists of only one word). Repeatedly, a linear programming problem is solved considering only, inequivalent distributions. A connection between this approach and we ighted coverings is also presented; the computations give new results for s uch coverings as a by-product.