On linear complementary-dual multinegacirculant codes

TL;DR

This paper characterizes and enumerates linear complementary-dual multinegacirculant codes, demonstrating their existence and properties, including infinite families with good relative distance, using algebraic and combinatorial methods.

Contribution

It provides algebraic characterization, enumeration, and asymptotic existence results for LCD multinegacirculant codes, especially for specific indices and co-indices.

Findings

Exact enumeration for indices 2 and 3

Infinite families with relative distance meeting a modified Varshamov-Gilbert bound

Algebraic characterization of LCD multinegacirculant codes

Abstract

Linear codes with complementary-duals (LCD) are linear codes that intersect with their dual trivially. Multinegacirculant codes of index $2$ that are LCD are characterized algebraically and some good codes are found in this family. Exact enumeration is performed for indices 2 and 3, and for all indices $t$ for a special case of the co-index by using their concatenated structure. Asymptotic existence results are derived for the special class of such codes that are one-generator and have co-index a power of two by means of Dickson polynomials. This shows that there are infinite families of LCD multinegacirculant codes with relative distance satisfying a modified Varshamov-Gilbert bound.