Improvement on Varshamov-Gilbert lower bound on minimum Hamming distance of linear codes

An improvement on the Varshamov-Gilbert lower bound on the minimum Hamming distance d of linear block codes is proposed. The improved bound is based on the assumption that, for an (n, k) block code, the number of distinct vectors resulting from the linear combination of every (d¿2) columns of the parity-check matrix is much less than the total number of vectors generated from such linear combinations. An expression for the largest possible number of distinct vectors obtainable for any (n, k) group code can therefore be introduced and shown to be a function of the weight distribution of the code.