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Cyclic codes over a non-commutative finite chain ring

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Abstract

In this study, we consider the finite (not necessary commutative) chain ring \(\mathcal {R}:=\mathbb {F}_{p^{m}}[u,\theta ]/{\left < u^{2} \right >}\), where θ is an automorphism of \(\mathbb {F}_{p^{m}}\), and completely explore the structure of left and right cyclic codes of any length N over \(\mathcal {R}\), that is, left and right ideals of the ring \(\mathcal {S}:=\mathcal {R}[x]/{\left < x^{N}-1 \right >}\). For a left (right) cyclic code, we determine the structure of its right (left) dual. Using the fact that self-dual codes are bimodules, we discuss on self-dual cyclic codes over \(\mathcal {R}\). Finally, we study Gray images of cyclic codes over \(\mathcal {R}\) and as some examples, three linear codes over \(\mathbb {F}_{4}\) with the parameters of the best known ones, but with different weight distributions, are obtained as the Gray images of cyclic codes over \(\mathcal {R}\).

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Acknowledgments

The author would like to thank anonymous referee for his (her) careful reading of this paper and invaluable comments. This research was in part supported by a grant from IPM (No. 94050080).

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Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, 81746-73441, Isfahan, Iran

    R. Sobhani

  2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), 19395-5746, Tehran, Iran

    R. Sobhani

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  1. R. Sobhani
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Sobhani, R. Cyclic codes over a non-commutative finite chain ring. Cryptogr. Commun. 10, 519–530 (2018). https://doi.org/10.1007/s12095-017-0238-5

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Mathematics Subject Classification (2010)