Half-cyclic, dihedral and half-dihedral codes
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Half-cyclic, dihedral and half-dihedral codes R. Jajcay1,2
· P. Potoˇcnik3,4 · S. Wilson5,6
Received: 11 February 2020 / Accepted: 4 June 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract This note discusses, in elementary terms, linear codes over Z2 which are closed under 2-step cyclic shifts, and classifies them in terms of special linear combinations of polynomials. Codes which are preserved under order-reversing automorphisms are also discussed, and a classification result in terms of special linear combinations of polynomials is obtained. Keywords Linear codes · Cyclic codes · Half-cyclic codes · Code symmetries Mathematics Subject Classification 94Bxx
The first author is supported by the Projects VEGA 1/0474/15, VEGA 1/0596/17, VEGA 1/0423/20, APVV-15-0220, and by the Slovenian Research Agency (research Projects N1-0038, N1-0062, J1-9108). The second author gratefully acknowledges the support of the Slovenian Research Agency, Programme P1–0294, and all three authors acknowledge the support of the bilateral Project BI-US/16-17-031.
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R. Jajcay [email protected] P. Potoˇcnik [email protected] S. Wilson [email protected]
1
Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia
2
Faculty of Mathematics, Natural Sciences and Information Technology, University of Primorska, Koper, Slovenia
3
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
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Institute of Mathematics, Physics, and Mechanics, Ljubljana, Slovenia
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Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ, USA
6
Faculty of Mathematics, Natural Sciences and Information Technology, University of Primorska, Koper, Slovenia
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R. Jajcay et al.
1 Introduction The results in this note were motivated by the study of the Praeger-Xu graphs, = PX(n, k) for k < n as defined in [5,10]. With three small exceptions, these graphs have symmetry groups Aut() isomorphic to Zn2 Dn . In investigating vertex-transitive subgroups of Aut(), in particular the Cayley subgroups, one needs to understand binary linear codes of length n which generalize the well-known cyclic codes. The more general codes needed in the study of the Praeger-Xu graphs are linear subspaces of Zn2 invariant under the actions of subgroups of the dihedral group Dn = R, M where R is the cyclic rotation of the positions (columns) of the code and M reverses the codewords. In this paper, we use the terms half-cyclic, dihedral and half-dihedral to indicate the three types of codes of interest. These are binary linear codes invariant under the action of R 2 , R, M, and R 2 , M, respectively. This paper’s primary results are Theorem 2.11 which classifies half-cyclic codes, Theorem 3.3 which classifies the dihedral codes, and Theorem 3.5 which classifies half-dihedral codes. It should be noted that half-cyclic codes constitute the simplest case of codes known under the name of quasi-cyclic codes. Quasi-cyclic codes
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