New extremal binary self-dual codes of length 68 via short kharaghani array over f_2 + uf_2

TL;DR

This paper introduces new construction methods for extremal binary self-dual codes of length 68 using short Kharaghani arrays over the ring F_2 + uF_2, leading to the discovery of previously unknown codes with specific weight enumerators.

Contribution

The authors develop novel construction techniques based on short Kharaghani arrays applicable over Frobenius rings, enabling the creation of 27 new extremal self-dual codes of length 68.

Findings

Constructed 27 new extremal binary self-dual codes of length 68.

Identified previously unknown weight enumerators for these codes.

Demonstrated applicability of the methods over the ring F_2 + uF_2.

Abstract

In this work, new construction methods for self-dual codes are given. The methods use the short Kharaghani array and a variation of it. These are applicable to any commutative Frobenius ring. We apply the constructions over the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various weight enumerators obtained as Gray images. By the use of an extension theorem for self-dual codes we were able to construct 27 new extremal binary self-dual codes of length 68. The existence of the extremal binary self-dual codes with these weight enumerators was previously unknown.