Near-Extremal Type I Self-Dual Codes with Minimal Shadow over GF(2) and GF(4)

Sunghyu Han

doi:10.3390/info9070172

Abstract

Binary self-dual codes and additive self-dual codes over GF(4) contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over GF(2) and GF(4) with minimal shadow. In particular, we prove that there is no near-extremal Type I [24m,12m,2m+2] binary self-dual code with minimal shadow if m≥323, and we prove that there is no near-extremal Type I (6m+1,26m+1,2m+1) additive self-dual code over GF(4) with minimal shadow if m≥22.

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
Munemasa studied the nonexistence of near-extremal Type I binary self-dual codes with minimal shadow [6]
N = 24m + 20 is the unique untouched code length for the nonexistence or an explicit bound for the length n of an extremal Type I binary self-dual code with minimal shadow

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. Munemasa studied the nonexistence of near-extremal Type I binary self-dual codes with minimal shadow [6]. We prove that there is no near-extremal Type I [24m, 12m, 2m + 2] binary self-dual code with minimal shadow if m ≥ 323. We prove that there is no near-extremal Type I (6m + 1, 26m+1 , 2m + 1) additive self-dual code over GF (4) with minimal shadow if m ≥ 22. We consider the nonexistence of extremal Type I binary self-dual codes with minimal shadow. We consider the nonexistence of extremal Type I additive self-dual codes over GF (4) with minimal shadow. All computer calculations in this study were performed using the mathematical software Maple

Extremal Type I Binary Self-Dual Codes with Minimal Shadow

Near-Extremal Type I Binary Self-Dual Codes with Minimal Shadow

The code C does not exist if:

Summary