Self-dual codes over GF(4)

F. J. MacWilliams; A. M. Odlyzko; N. J.A. Sloane; H. N. Ward

doi:10.1016/0097-3165(78)90021-3

Self-dual codes over GF(4)

F. J. MacWilliams, A. M. Odlyzko, N. J.A. Sloane, H. N. Ward

School of Mathematics

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102 Scopus citations

Abstract

This paper studies codes C over GF(4) which have even weights and have the same weight distribution as the dual code C^⊥. Some of the results are as follows. All such codes satisfy C^⊥ = C (If C^⊥= C, T has a binary basis.) The number of such C's is determined, and those of length ≤14 are completely classified. The weight enumerator of C is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for C to be extended cyclic. Two new 5-designs are constructed. A generator matrix for C can be taken to have the form [I | B], where B^⊥ = B. We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.

Original language	English (US)
Pages (from-to)	288-318
Number of pages	31
Journal	Journal of Combinatorial Theory, Series A
Volume	25
Issue number	3
DOIs	https://doi.org/10.1016/0097-3165(78)90021-3
State	Published - Nov 1978

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title = "Self-dual codes over GF(4)",

abstract = "This paper studies codes C over GF(4) which have even weights and have the same weight distribution as the dual code C⊥. Some of the results are as follows. All such codes satisfy C⊥ = C (If C⊥= C, T has a binary basis.) The number of such C's is determined, and those of length ≤14 are completely classified. The weight enumerator of C is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for C to be extended cyclic. Two new 5-designs are constructed. A generator matrix for C can be taken to have the form [I | B], where B⊥ = B. We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.",

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year = "1978",

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T1 - Self-dual codes over GF(4)

AU - MacWilliams, F. J.

AU - Odlyzko, A. M.

AU - Sloane, N. J.A.

AU - Ward, H. N.

PY - 1978/11

Y1 - 1978/11

N2 - This paper studies codes C over GF(4) which have even weights and have the same weight distribution as the dual code C⊥. Some of the results are as follows. All such codes satisfy C⊥ = C (If C⊥= C, T has a binary basis.) The number of such C's is determined, and those of length ≤14 are completely classified. The weight enumerator of C is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for C to be extended cyclic. Two new 5-designs are constructed. A generator matrix for C can be taken to have the form [I | B], where B⊥ = B. We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.

AB - This paper studies codes C over GF(4) which have even weights and have the same weight distribution as the dual code C⊥. Some of the results are as follows. All such codes satisfy C⊥ = C (If C⊥= C, T has a binary basis.) The number of such C's is determined, and those of length ≤14 are completely classified. The weight enumerator of C is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for C to be extended cyclic. Two new 5-designs are constructed. A generator matrix for C can be taken to have the form [I | B], where B⊥ = B. We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.

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Self-dual codes over GF(4)

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