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論不於GF(16)的加法碼

陳瑋琪

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Abstract In this paper we generalize Gaborit's paper iOn Additive GF(4) Codesi[2].The additive GF(4) codes are of interest because of their connection to quantum codes [1]. First we de ne the conjugates of nite eld of 16 elements. Next we give the de nition of additive GF(16) codes, the trace map, the trace inner product and their properties, and example of additive GF(16) code. As usual we de ne dual, self-orthogonal, self-dual codes of additive GF(16) codes. Finally, the de nition equivalent codes and automorphism group of additive GF(16) codes are given.

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Abstract In this paper we generalize Gaborit's paper iOn Additive GF(4) Codesi[2].The additive GF(4) codes are of interest because of their connection to quantum codes [1]. First we de ne the conjugates of nite eld of 16 elements. Next we give the de nition of additive GF(16) codes, the trace map, the trace inner product and their properties, and example of additive GF(16) code. As usual we de ne dual, self-orthogonal, self-dual codes of additive GF(16) codes. Finally, the de nition equivalent codes and automorphism group of additive GF(16) codes are given.
Additive code C over GF(4) of length n is an additive subgroup of GF(4)n. It is well known [4] that the problem of finding stabilizer quantum error-correcting codes is transformed into problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to construct good additive self-dual codes of length 13 ≤ n ≤ 21. In this paper we classify all extremal (optimal) codes of lengths 13 and 14, and we construct many extremal codes of lengths 15 and 16. Also, we construct some new extremal codes of lengths 17,18,19, and 21. We give the current status of known extremal (optimal) additive self-dual codes of lengths 13 to 21.
. In this paper we discuss the status of the classication of the extremal additive codes over GF (4) which are self-dual under the trace inner product. These codes are of interest because of their connection to quantum codes. There are 5 inequivalent codes of length 8, 8 of length 9, at least 56 of length 10, and exactly 1 of length 11. We discuss the methods of classication, the concepts of lengthening and shortening, and connections to binary codes. 1. Introduction An additive code C over GF (4) of length n is an additive subgroup of GF (4) n . As C is a free GF(2)-module, it has size 2 k for some 0 k 2n. We call C an (n; 2 k ) code. It has a basis, as a GF(2)-module, consisting of k basis vectors; a generator matrix of C will be a k n matrix with entries in GF (4) whose rows are a basis of C. Interest in additive codes over GF (4) has arisen because of their correspondence to quantum codes as described in [3]. There is a natural inner product arising from the trace m...
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 Hermitian Self-Dual Additive Codes Over GF(9){\rm GF}(9)GF(9)
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 lengths 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 lengths 11 and 12. There are 56 005 876 (11, 311, 5) codes and 6493 (12, 312, 6) codes. We also find the smallest codes with trivial automorphism group.