On the classification of Hermitian self-dual additive codes over GF(9)

TL;DR

This paper extends the classification of Hermitian self-dual additive codes over GF(9) to lengths 9 and 10 using a novel graph-based algorithm, revealing millions of codes and their automorphism properties.

Contribution

It introduces a new graph-based algorithm for classifying Hermitian self-dual additive codes over GF(9), extending previous classifications to longer code lengths.

Findings

Classified all codes of length 9 and 10 over GF(9).

Identified over 56 million codes of length 11 and 12.

Discovered the smallest codes with trivial automorphism groups.

Abstract

Additive codes over GF(9) that are self-dual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum error-correcting codes. However, these codes have so far received far less interest from coding theorists than self-dual additive codes over GF(4), which correspond to binary quantum codes. Self-dual additive codes over GF(9) have been classified up to length 8, and in this paper we extend the complete classification to codes of length 9 and 10. The classification is obtained by using a new algorithm that combines two graph representations of self-dual additive codes. The search space is first reduced by the fact that every code can be mapped to a weighted graph, and a different graph is then introduced that transforms the problem of code equivalence into a problem of graph isomorphism. By an extension technique, we are able to classify all optimal codes of length 11 and 12. There are 56,005,876 (11,3^11,5) codes and 6493 (12,3^12,6) codes. We also find the smallest codes with trivial automorphism group.