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November  2009, 3(4): 329-348. doi: 10.3934/amc.2009.3.329

Graph-based classification of self-dual additive codes over finite fields

Lars Eirik Danielsen 1,

1. 

Department of Informatics, University of Bergen, PO Box 7803, N-5020 Bergen, Norway

Received  February 2009 Revised  September 2009 Published  November 2009

Quantum stabilizer states over Fm can be represented as self-dual additive codes over Fm2. 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 F4. In this paper we classify selfdual additive codes over F9, F16, and F25. Assuming that the classical MDS conjecture holds, we are able to classify all self-dual additive MDS codes over F9 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.
Keywords: quantum codes, local complementation, circulant codes, self-dual codes, Additive codes, graphs, MDS codes., stabilizer states.
Mathematics Subject Classification: Primary: 94B05, 05C30; Secondary: 94B60, 05C5.
Citation: Lars Eirik Danielsen. Graph-based classification of self-dual additive codes over finite fields. Advances in Mathematics of Communications, 2009, 3 (4) : 329-348. doi: 10.3934/amc.2009.3.329
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