On the Number of Minimum-Weight Words in Block Codes

Lower bounds on the number of minimum distances are considered for arbitrary block codes of given length, size, and base. These are lower bounds on the number of minimal-weight words for distance-invariant codes containing the null word. We apply the notion of uniformly packed code and MacWilliams's transform of the distance distribution of a code to derive the conditions when these bounds are attainable. Examples of codes attaining this bound are given.