Computational Results of Duadic Double Circulant Codes

TL;DR

This paper introduces duadic double circulant codes, a new subclass of algebraic codes that systematically generate optimal and best-known parameter codes over various finite fields.

Contribution

It defines duadic double circulant codes, a generalization of quadratic double circulant codes, and provides a construction method using 4-cyclotomic cosets.

Findings

Discovery of a new ternary self-dual [76,38,18] code.

Rediscovery of optimal binary self-dual codes with specific parameters.

Construction of codes over multiple finite fields with optimal or best-known parameters.

Abstract

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called {\em{duadic double circulant codes}}, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb F_2, \mathbb F_3, \mathbb F_4, \mathbb F_5$, and $\mathbb F_7$. In particular, we find a new ternary self-dual $[76,38,18]$ code and easily rediscover optimal binary self-dual codes with parameters $[66,33,12]$, $[68,34,12]$, $[86,43,16]$, and $[88,44,16]$ as well as a formally self-dual binary $[82,41,14]$ code.