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

Abstract

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 operationcorrespond 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 MDSconjecture holds, we are able to classify all self-dual additive MDS codes over F9 by using an extension technique. We prove that the minimum distanceof a self-dual additive code is related to the minimum vertex degree in theassociated graph orbit. Circulant graph codes are introduced, and a computersearch reveals that this set contains many strong codes. We show that some ofthese codes have highly regular graph representations.