Weight Distributions of Hamming Codes

TL;DR

This paper derives a recursive formula for the weight distribution of Hamming codes over finite fields, using Pless power moments and exponential sums, under specific conditions on parameters.

Contribution

It introduces a new recursive formula for the weight distribution of Hamming codes when (m, q-1)=1, combining Pless power moments with exponential sum techniques.

Findings

Recursive formula for weight distribution derived

Applicable for Hamming codes with (m, q-1)=1

Uses Pless power moments and exponential sums

Abstract

We derive a recursive formula determing the weight distribution of the [n=(q^m-1)/(q-1), n-m, 3] Hamming code H(m,q), when (m, q-1)=1. Here q is a prime power. The proof is based on Moisio's idea of using Pless power moment identity together with exponential sum techniques.