On the Weight Distribution of the Extended Quadratic Residue Code of Prime 137

TL;DR

This paper determines the weight distribution of the extended quadratic residue code for prime 137 using Gleason's theorem, avoiding exhaustive enumeration, and verifies the distribution through congruences.

Contribution

It provides a novel method to compute the weight distribution of the prime 137 code without exhaustive enumeration, extending known results.

Findings

Weight distribution for prime 137 code is fully determined.

A new approach avoids exhaustive codeword enumeration.

Distributions are verified via congruences.

Abstract

The Hamming weight enumerator function of the formally self-dual even, binary extended quadratic residue code of prime p = 8m + 1 is given by Gleason's theorem for singly-even code. Using this theorem, the Hamming weight distribution of the extended quadratic residue is completely determined once the number of codewords of Hamming weight j A_j, for 0 <= j <= 2m, are known. The smallest prime for which the Hamming weight distribution of the corresponding extended quadratic residue code is unknown is 137. It is shown in this paper that, for p=137 A_2m = A_34 may be obtained with out the need of exhaustive codeword enumeration. After the remainder of A_j required by Gleason's theorem are computed and independently verified using their congruences, the Hamming weight distributions of the binary augmented and extended quadratic residue codes of prime 137 are derived.