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Higher weights for codes over rings

Abstract

A generalized definition of higher weights for codes over finite chain rings and principal ideal rings and bounds on the minimum higher weights in this setting are given. Using this we generalize the definition for higher MDS and MDR codes. Computationally, the higher weight enumerator of lifted Hamming and Simplex codes over \({\mathbb{Z}_4}\), the minimum higher weights for the lifted code of the binary [8,4,4] self-dual extended Hamming code, the lifted code of the ternary [12,6,6] self-dual Golay code and the lifted code of the binary [24,12,8] self-dual Golay code are given. Joint weight enumerators are used to produce MacWilliams relations for specific higher weight enumerators.

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Correspondence to Steven T. Dougherty.

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The author H. Liu is supported by the National Natural Science Foundation of China (10871079).

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Dougherty, S.T., Han, S. & Liu, H. Higher weights for codes over rings. AAECC 22, 113–135 (2011). https://doi.org/10.1007/s00200-011-0140-x

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  • DOI: https://doi.org/10.1007/s00200-011-0140-x

Keywords

  • Higher weights
  • Finite chain rings
  • Finite principal ideal rings