Graph-Based Classification of Self-Dual Additive Codes over Finite Fields

TL;DR

This paper classifies self-dual additive codes over GF(9), GF(16), and GF(25) using graph representations, extending previous work on GF(4), and explores their properties and strong code examples.

Contribution

It extends the classification of self-dual additive codes to new finite fields and introduces circulant graph codes with highly regular structures.

Findings

Classified codes over GF(9), GF(16), GF(25)

Established relation between code distance and graph vertex degree

Identified strong codes with regular graph representations

Abstract

Quantum stabilizer states over GF(m) can be represented as self-dual additive codes over GF(m^2). These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used this fact to classify self-dual additive codes over GF(4). In this paper we classify self-dual additive codes over GF(9), GF(16), and GF(25). Assuming that the classical MDS conjecture holds, we are able to classify all self-dual additive MDS codes over GF(9) by using an extension technique. We prove that the minimum distance of a self-dual additive code is related to the minimum vertex degree in the associated graph orbit. Circulant graph codes are introduced, and a computer search reveals that this set contains many strong codes. We show that some of these codes have highly regular graph representations.