New extremal binary self-dual codes of lengths 66 and 68 from codes over r_k,m

TL;DR

This paper constructs new extremal binary self-dual codes of lengths 66 and 68 using novel algebraic methods over rings, expanding the known catalog of such codes with previously unreported weight enumerators.

Contribution

It introduces new extremal binary self-dual codes of lengths 66 and 68 derived from codes over rings R_k,m, including the first known codes with specific weight enumerators.

Findings

11 new codes of length 66

39 new codes of length 68

First construction of codes with certain weight enumerators

Abstract

In this work, four circulant and quadratic double circulant (QDC) constructions are applied to the family of the rings R_k,m. Self-dual binary codes are obtained as the Gray images of self-dual QDC codes over R_k,m. Extremal binary self-dual codes of length 64 are obtained as Gray images of ?-four circulant codes over R_2,1 and R_2,2. Extremal binary self-dual codes of lengths 66 and 68 are constructed by applying extension theorems to the F_2 and R_2,1 images of these codes. More precisely, 11 new codes of length 66 and 39 new codes of length 68 are discovered. The codes with these weight enumerators are constructed for the first time in literature. The results are tabulated.