Additive Self-Dual Codes over GF(4) with Minimal Shadow

Sunghyu Han

doi:10.3390/info9040081

Abstract

We define additive self-dual codes over G F ( 4 ) with minimal shadow, and we prove the nonexistence of extremal Type I additive self-dual codes over G F ( 4 ) with minimal shadow for some parameters.

Highlights

There are many interesting classes of codes in coding theory such as cyclic codes, quadratic residue codes, algebraic geometry codes, and self-dual codes
We prove the nonexistence of extremal Type I codes with minimal shadow for some parameters
We investigate Type I additive self-dual codes over GF (4)

Summary

Introduction

There are many interesting classes of codes in coding theory such as cyclic codes, quadratic residue codes, algebraic geometry codes, and self-dual codes. If the code length is large, determining the highest minimum weight of self-dual codes becomes difficult. Conway and Sloane gave an upper bound on the minimum weight of binary self-dual codes [2]. They used the concept of shadow codes. Bouyuklieva and Willems studied binary self-dual codes for which the minimum weight of the shadow codes had the smallest possible value. They proved that extremal binary self-dual codes with minimal shadow for particular parameters do not exist [4].

Preliminaries

Summary