On the Classification of Hermitian Self-Dual Additive Codes over GF(9) (2011)
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BibTeX
@MISC{Danielsen11onthe,
author = {Lars Eirik Danielsen},
title = { On the Classification of Hermitian Self-Dual Additive Codes over GF(9)},
year = {2011}
}
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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.
Keyphrases
hermitian self-dual additive code self-dual additive code trivial automorphism group new algorithm search space weighted graph different graph graph isomorphism quantum code natural application ternary quantum error-correcting code additive code code equivalence extension technique hermitian trace inner product complete classification optimal code quantum information theory graph representation
