New extremal binary self-dual codes of length 68 from quadratic residue codes over f_2+uf_2+u^2f_2

TL;DR

This paper constructs new extremal binary self-dual codes of length 68 using quadratic residue codes over a specific ring and Gray maps, significantly expanding known code families.

Contribution

It introduces novel constructions of extremal self-dual codes via quadratic residue codes over a ring and Gray maps, resulting in 32 new codes of length 68.

Findings

32 new extremal binary self-dual codes of length 68

363 Type I codes of parameters [72; 36; 12]

A Type II code with parameters [96; 48; 16]

Abstract

In this work, quadratic reside codes over the ring F2 +uF2 +u^2F2 with u^3 = u are considered. A duality and distance preserving Gray map from F2 + uF2 + u^2F2 to (F_2)^3 is defined. By using quadratic double circulant, quadratic bordered double circulant constructions and their extensions self- dual codes of different lengths are obtained. As Gray images of these codes and their extensions, a substantial number of new extremal self-dual binary codes are found. More precisely, thirty two new extremal binary self-dual codes of length 68, 363 Type I codes of parameters [72; 36; 12], a Type II [72; 36; 12] code and a Type II [96; 48; 16] code with new weight enumerators are obtained through these constructions. The results are tabulated.