Enumeration of Self-Dual Codes of Length 6 over ℤp

Whan-Hyuk Choi

doi:10.3390/math7100882

Abstract

The purpose of this paper is to classify and enumerate self-dual codes of length 6 over finite field Z p . First, we classify these codes into three cases: decomposable, indecomposable non-MDS and MDS codes. Then, we complete the classification of non-MDS self-dual codes of length 6 over Z p for all primes p in terms of their automorphism group. We obtain all inequivalent classes and find the necessary and sufficient conditions for the existence of each class. Finally, we obtain the number of MDS self-dual codes of length 6.

Highlights

The classification problem is one of the fundamental problems in various areas of mathematics.From the time self-dual codes began to attract attention amongst coding theorists, many papers have been published to classify binary self-dual codes [1,2,3] and non-binary self-dual codes [4,5,6,7,8].While self-dual codes of moderate lengths are classified over a finite field Z p for a fixed prime p in these papers, the first efforts are made in [9] to classify self-dual codes of the fixed length 4 over Z p for all primes p
In [9], it is shown that the classification of self-dual codes over Z p for all primes p is essential to classify self-dual codes over integer ring Zm for arbitrary m
With these results, we calculate the exact number of distinct MDS self-dual codes of length 6, so that we can obtain the mass formula for MDS codes

Summary

Introduction

The classification problem is one of the fundamental problems in various areas of mathematics. Calculating the total number of codes and automorphisms of each equivalence class is critical for the classification of self-dual codes. It is shown that self-dual codes of length 6 are classified into the following three cases by checking the number of zero elements of the standard generator matrix: decomposable case, indecomposable non-MDS case, and MDS case. For the indecomposable non-MDS codes, we complete the classification by computing the total number of bisorted standard generator matrices and orbits of each equivalent class. With these results, we calculate the exact number of distinct MDS self-dual codes of length 6, so that we can obtain the mass formula for MDS codes.

Preliminaries

Self-Dual Codes of Length 6 over Z p

Automorphism of Self-Dual Codes of Length 6 over Z p

Enumeration of Self-Dual Codes of Length 6 over Z p