On the classification of Hermitian selfdual additive codes over GF(9)
Abstract
Additive codes over GF(9) that are selfdual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum errorcorrecting codes. However, these codes have so far received far less interest from coding theorists than selfdual additive codes over GF(4), which correspond to binary quantum codes. Selfdual 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 selfdual 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2011
 arXiv:
 arXiv:1106.2428
 Bibcode:
 2011arXiv1106.2428E
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Information Theory;
 Quantum Physics
 EPrint:
 12 pages, 6 figures