Directed Graph Representation of Half-Rate Additive Codes over GF(4)

TL;DR

This paper introduces a directed graph representation for (n,2^n) additive codes over GF(4), enabling classification and construction of high-quality codes, including new near-extremal and better minimum distance codes.

Contribution

It generalizes graph representations to directed graphs for additive codes over GF(4), facilitating classification and new code constructions.

Findings

Classified additive codes up to length 7 using directed graphs.

Constructed many high minimum distance codes, including new near-extremal codes.

Produced codes with better minimum distance than known self-dual codes.

Abstract

We show that (n,2^n) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n,2^n) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.