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Semisimple multivariable \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^l}\)

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Abstract

Let \(\mathbb {F}_q\) be a finite field of cardinality \(q\), \(l\) a prime number and \(\mathbb {F}_{q^l}\) an extension field of \(\mathbb {F}_q\) with degree \(l\). The structure and canonical form decompositions of semisimple multivariable \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^l}\) are presented. Enumeration and construction of these codes are then investigated. Especially, dual codes, self-orthogonality and self-duality of semisimple abelian \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^l}\) are studied. Furthermore, self-dual and \(\gamma \)-self dual semisimple abelian \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^2}\) are considered.

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Acknowledgments

Part of this work was done when Y. Cao was visiting the Chern Institute of Mathematics, Nankai University, Tianjin, China. Y. Cao would like to thank the institution for the kind hospitality. The research is supported by the National Key Basic Research Program of China (Grant No. 2013CB834204), and the National Natural Science Foundation of China (Grant Nos. 10971160, 61171082 and 11201269).

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Authors and Affiliations

  1. School of Sciences, Shandong University of Technology, Zibo , 255091, Shandong, China

    Yonglin Cao

  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin , 300071, China

    Jian Gao & Fang-Wei Fu

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  1. Yonglin Cao
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  2. Jian Gao
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  3. Fang-Wei Fu
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Correspondence to Yonglin Cao.

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Communicated by A. Winterhof.

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Cao, Y., Gao, J. & Fu, FW. Semisimple multivariable \(\mathbb {F}_q\)-linear codes over \(\mathbb {F}_{q^l}\) . Des. Codes Cryptogr. 77, 153–177 (2015). https://doi.org/10.1007/s10623-014-9994-9

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  • DOI: https://doi.org/10.1007/s10623-014-9994-9

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