Double and bordered \$\alpha\$-circulant self-dual codes over finite commutative chain rings

TL;DR

This paper develops a method for lifting self-dual -circulant codes over finite chain rings, preserving self-duality, and applies it to classify and optimize certain Z_4-linear codes up to length 64.

Contribution

It introduces a novel lifting technique for self-dual -circulant codes over finite chain rings, ensuring self-duality is maintained during the process.

Findings

Identified optimal minimum Lee distances for codes up to length 64.

Developed a linear system approach for preserving self-duality during lifting.

Applied the method to classify Z_4-linear double nega-circulant and bordered circulant codes.

Abstract

In this paper we investigate codes over finite commutative rings R, whose generator matrices are built from \$\alpha\$-circulant matrices. For a non-trivial ideal I<R we give a method to lift such codes over R/I to codes over R, such that some isomorphic copies are avoided. For the case where I is the minimal ideal of a finite chain ring we refine this lifting method: We impose the additional restriction that lifting preserves self-duality. It will be shown that this can be achieved by solving a linear system of equations over a finite field. Finally we apply this technique to Z_4-linear double nega-circulant and bordered circulant self-dual codes. We determine the best minimum Lee distance of these codes up to length 64.