On the average Hamming distance for binary codes

By using the dual distance distribution and its properties for binary code C with length n and size M, the Althöfer-Sillke inequality is improved for odd M. Let β(n, M) denote the minimum value of average Hamming distance (AHD) of binary (n, M) codes. In this paper, β(n, 2n − 1), β(n, 2n−1 − 1) and β(n, 2n−1 + 1) are determined. Two recursive inequalities for β(n, M) are derived. Furthermore, the variance of AHD of code C is studied, and lower and upper bounds are presented.