Some Constant Weight Codes from Primitive Permutation Groups

Derek H Smith,Roberto Montemanni

doi:10.37236/2702

Abstract

In recent years the detailed study of the construction of constant weight codes has been extended from length at most 28 to lengths less than 64. Andries Brouwer maintains web pages with tables of the best known constant weight codes of these lengths. In many cases the codes have more codewords than the best code in the literature, and are not particularly easy to improve. Many of the codes are constructed using a specified permutation group as automorphism group. The groups used include cyclic, quasi-cyclic, affine general linear groups etc. sometimes with fixed points. The precise rationale for the choice of groups is not clear.In this paper the choice of groups is made systematic by the use of the classification of primitive permutation groups. Together with several improved techniques for finding a maximum clique, this has led to the construction of 39 improved constant weight codes.

Highlights

In recent years the detailed study of the construction of constant weight codes has been extended from length at most 28 to lengths less than 64
A constant weight binary code is a set of binary vectors of length n, weight w and minimum Hamming distance d
The maximum possible number of vectors in a constant weight code is usually referred to as A(n, d, w). These codes have an important role in the theory of error-correcting codes [14]. They have been used in applications such as the design of demultiplexers for nano-scale memories [13] and the construction of frequency hopping lists for use in GSM networks [16]

Summary

Introduction

A constant weight binary code is a set of binary vectors of length n, weight w and minimum Hamming distance d. The maximum possible number of vectors in a constant weight code is usually referred to as A(n, d, w). These codes have an important role in the theory of error-correcting codes [14]. A variety of methods for obtaining constructive lower bounds for A(n, d, w) can be found in [3], where tables of best known codes are given for n 28. A maximum weighted clique of Γ(n, d, w) (complete subgraph with the maximum sum of vertex weights) corresponds to the largest code C(n, d, w) obtainable by this method from the group G. When a new best code is found it may give further new best codes by shortening

Primitive Permutation Groups

Maximum Clique Algorithms

Heuristic Post-processing

Shortening

The Constant Weight Codes Constructed

Conclusion