On the Existence of Certain Optimal Self-Dual Codes with Lengths Between 74 and 116

Tao Zhang,Jerod Michel,Gennian Ge,Tao Feng

doi:10.37236/5213

Abstract

The existence of optimal binary self-dual codes is a long-standing research problem. In this paper, we present some results concerning the decomposition of binary self-dual codes with a dihedral automorphism group $D_{2p}$, where $p$ is a prime. These results are applied to construct new self-dual codes with length $78$ or $116$. We obtain $16$ inequivalent self-dual $[78,39,14]$ codes, four of which have new weight enumerators. We also show that there are at least $141$ inequivalent self-dual $[116,58,18]$ codes, most of which are new up to equivalence. Meanwhile, we give some restrictions on the weight enumerators of singly even self-dual codes. We use these restrictions to exclude some possible weight enumerators of self-dual codes with lengths $74$, $76$, $82$, $98$ and $100$.

Highlights

Binary self-dual codes have been of particular interest for some time
We present some results concerning the decomposition of binary self-dual codes with a dihedral automorphism group D2p, where p is a prime
We investigate binary self-dual codes with a dihedral automorphism group D2p of order 2p, where p is an odd prime

Summary

Introduction

Binary self-dual codes have been of particular interest for some time now. The extended Hamming [8, 4, 4] code, the extended Golay [24, 12, 8] code and certain extended quadratic. In 2004, Dontcheva et al [17] extended the results to the decomposition of binary self-dual codes possessing an automorphism of order pq, where p and q are odd prime numbers This technique yields many extremal or optimal codes which possess an automorphism (see [9], [10], [11], [15], [39], [41], [40], [42]). The results will be applied to classify all binary self-dual [78, 39, 14] codes with a dihedral automorphism group D38 Some of these have weight enumerator of form W78,1 with α = 0 and β = −38 (the existence of such codes was previously unknown). We extend these results to the case where the codes have dihedral automorphism group D2p.

Preliminaries

New Optimal Self-Dual Codes with Dihedral Automorphism Group D2p

Some Restrictions on Weight Enumerators

Conclusion