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Directed Graph Representation of HalfRate Additive Codes over GF(4)
"... Abstract. We show that (n, 2 n, d) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on selfdual additive codes over GF(4), which correspond to undirected graphs. Graph representation greatly reduces the complexity of code classification, and enables ..."
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Abstract. We show that (n, 2 n, d) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on selfdual additive codes over GF(4), which correspond to undirected graphs. Graph representation greatly reduces the complexity of code classification, and enables us to classify additive (n, 2 n, d) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally selfdual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new nearextremal formally selfdual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known selfdual codes. 1
Interlace Polynomials: Enumeration, Unimodality, and Connections to Codes
, 2008
"... The interlace polynomial q was introduced by Arratia, Bollobás, and Sorkin. It encodes many properties of the orbit of a graph under edge local complementation (ELC). The interlace polynomial Q, introduced by Aigner and van der Holst, similarly contains information about the orbit of a graph under l ..."
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The interlace polynomial q was introduced by Arratia, Bollobás, and Sorkin. It encodes many properties of the orbit of a graph under edge local complementation (ELC). The interlace polynomial Q, introduced by Aigner and van der Holst, similarly contains information about the orbit of a graph under local complementation (LC). We have previously classified LC and ELC orbits, and now give an enumeration of the corresponding interlace polynomials of all graphs of order up to 12. An enumeration of all circle graphs of order up to 12 is also given. We show that there exist graphs of all orders greater than 9 with interlace polynomials q whose coefficient sequences are nonunimodal, thereby disproving a conjecture by Arratia et al. We have verified that for graphs of order up to 12, all polynomials Q have unimodal coefficients. It has been shown that LC and ELC orbits of graphs correspond to equivalence classes of certain errorcorrecting codes and quantum states. We show that the properties of these codes and quantum states are related to properties of the associated interlace polynomials. 1
Additive circulant graph codes over GF (4)
"... Abstract. In this paper we consider additive circulant graph codes over GF(4) and an algorithm for their construction. Also, we present some new results obtained by this algorithm. 1 ..."
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Abstract. In this paper we consider additive circulant graph codes over GF(4) and an algorithm for their construction. Also, we present some new results obtained by this algorithm. 1
On the Classification of Hermitian SelfDual Additive Codes over GF(9)
, 2011
"... 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 ..."
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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.
On the Classification of SelfDual Additive Codes over GF(9)
"... Abstract—Additive codes over GF(9) that are selfdual with respect to the Hermitian trace inner product have previously been classified up to length 8. In this paper, all codes of length 9 and 10 are classified, using a new algorithm that combines two graph representations of codes. First, the searc ..."
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Abstract—Additive codes over GF(9) that are selfdual with respect to the Hermitian trace inner product have previously been classified up to length 8. In this paper, all codes of length 9 and 10 are classified, using a new algorithm that combines two graph representations of codes. First, the search space is reduced by the fact that every selfdual additive code can be mapped to a weighted graph. Then a different graph is described that transforms the problem of code equivalence into a problem of graph isomorphism. I.
ON THE THEORY OF FqLINEAR Fq tCODES
, 2013
"... In [7], selforthogonal additive codes over F4 under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called Fqlinear F q tcodes. We examine a number o ..."
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In [7], selforthogonal additive codes over F4 under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called Fqlinear F q tcodes. We examine a number of classical results from the theory of Fqlinear codes, and see how they must be modified to give analogous results for Fqlinear F q tcodes. Included in the topics examined are the MacWilliams Identities, the Gleason polynomials, the GleasonPierce Theorem, Mass Formulas, the Balance Principle, the Singleton Bound, and MDS codes. We also classify certain of these codes for small lengths using the theory developed.