An improved upper bound on self-dual codes over finite fields $GF(11), GF(19)$, and $GF(23)$

TL;DR

This paper introduces new construction methods for symmetric self-dual codes over GF(11), GF(19), and GF(23), leading to the discovery of 153 new codes and improved bounds on their minimum distance.

Contribution

It presents novel construction techniques for symmetric self-dual codes over specific finite fields, resulting in numerous new codes and tighter bounds on their parameters.

Findings

Constructed symmetric self-dual codes of length less than 42 over GF(11), GF(19), GF(23)

Discovered 153 new self-dual codes with improved parameters

Enhanced bounds on the maximum minimum distance of self-dual codes

Abstract

This paper gives new methods of constructing {\it symmetric self-dual codes} over a finite field $GF(q)$ where $q$ is a power of an odd prime. These methods are motivated by the well-known Pless symmetry codes and quadratic double circulant codes. Using these methods, we construct an amount of symmetric self-dual codes over $GF(11)$, $GF(19)$, and $GF(23)$ of every length less than 42. We also find 153 {\it new} self-dual codes up to equivalence: they are $[32, 16, 12]$, $[36, 18, 13]$, and $[40, 20,14]$ codes over $GF(11)$, $[36, 18, 14]$ and $[40, 20, 15]$ codes over $GF(19)$, and $[32, 16, 12]$, $[36, 18, 14]$, and $[40, 20, 15]$ codes over $GF(23)$. They all have new parameters with respect to self-dual codes. Consequently, we improve bounds on the highest minimum distance of self-dual codes, which have not been significantly updated for almost two decades.