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Directed Graph Representation of Half-Rate 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 self-dual 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 self-dual 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 self-dual 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 near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual 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 non-unimodal, 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 error-correcting 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 Self-Dual Additive Codes over GF(9)
, 2011
"... 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 ..."
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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.
On the Classification of Self-Dual Additive Codes over GF(9)
"... Abstract—Additive codes over GF(9) that are self-dual 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 self-dual 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 self-dual 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 Fq-LINEAR Fq t-CODES
, 2013
"... In [7], self-orthogonal 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 Fq-linear F q t-codes. We examine a number o ..."
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In [7], self-orthogonal 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 Fq-linear F q t-codes. We examine a number of classical results from the theory of Fq-linear codes, and see how they must be modified to give analogous results for Fq-linear F q t-codes. Included in the topics examined are the MacWilliams Identities, the Gleason polynomials, the Gleason-Pierce 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.
