Weight Distributions of Hamming Codes (II)

TL;DR

This paper extends the understanding of Hamming code weight distributions by removing previous restrictions, providing a more general recursive formula applicable to all positive integers m and prime powers q.

Contribution

It generalizes the recursive formula for Hamming code weight distributions to all positive integers m and prime powers q, removing earlier restrictions.

Findings

The recursive formula holds for all positive integers m and prime powers q.

The previous restriction (m, q-1)=1 is unnecessary for the formula's validity.

The general formula simplifies analysis of Hamming codes across broader parameters.

Abstract

In a previous paper, we derived a recursive formula determining the weight distributions of the [n=(q^m-1)/(q-1)] Hamming code H(m,q), when (m,q-1)=1. Here q is a prime power. We note here that the formula actually holds for any positive integer m and any prime power q, without the restriction (m, q-1)=1.