Abstract
For codes, there are multiple notions of isomorphism. For example, we can consider isomorphisms that only permute the coordinates of codewords, or isomorphisms that not only permute the coordinates of codewords but also multiply each coordinate by a scalar (not necessarily the same scalar for each coordinate) as it permutes the coordinates. Isomorphisms of cyclic codes of the first kind have been studied in some circumstances—we will call them permutation isomorphisms—and our purpose is to begin the study of the second kind of isomorphism—which we call monomial isomorphisms—for cyclic codes. We give examples of cyclic codes that are monomially isomorphic but not permutationally isomorphic. We also show that the monomial isomorphism problem for cyclic codes of length \(n\) over \({\mathbb F}_q\) reduces to the permutation isomorphism problem for cyclic codes of length \(n\) over \({\mathbb F}_q\) if and only if \(\mathrm{gcd}(n,q-1) = 1\). Applying known results, this solves the monomial isomorphism problem for cyclic codes satisfying \(\mathrm{gcd}(n,q(q-1)) = 1\). Additionally, we solve the monomial isomorphism problem for cyclic codes of prime length over all finite fields. Finally, our results also hold for some codes that are not cyclic.
Similar content being viewed by others
References
Babai L.: Isomorphism problem for a class of point-symmetric structures. Acta Math. Acad. Sci. Hung. 29(3–4), 329–336 (1977).
Dixon J.D., Mortimer B.: Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, New York (1996).
Dobson E.: On the Cayley isomorphism problem for Cayley objects of nilpotent groups of some orders (submitted).
Dobson E.: Isomorphism problem for Cayley graphs of \({\mathbb{Z}}_{p}^{3}\). Discret. Math. 147(1–3), 87–94 (1995).
Dobson E.: Classification of vertex-transitive graphs of order a prime cubed. I. Discret. Math. 224(1–3), 99–106 (2000).
Dobson E.: On isomorphisms of abelian Cayley objects of certain orders. Discret. Math. 266(1–3), 203–215 (2003) The 18th British Combinatorial Conference (Brighton, 2001).
Filip P., Heise W.: Monomial Code-Isomorphisms, Combinatorics ’84 (Bari, 1984). North-Holland Mathematics Study vol. 123, pp. 217–223, North-Holland, Amsterdam (1986).
Godsil C.D.: On Cayley graph isomorphisms. Ars Comb. 15, 231–246 (1983).
Hirasaka M., Muzychuk M.: An elementary abelian group of rank 4 is a CI-group. J. Comb. Theory Ser. A 94(2), 339–362 (2001).
Huffman W.C., Job V., Pless V.: Multipliers and generalized multipliers of cyclic objects and cyclic codes. J. Comb. Theory Ser. A 62(2), 183–215 (1993).
Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge (2003).
Hungerford T.W.: Algebra. Graduate Texts in Mathematics, vol. 73, Springer, New York (1980); reprint of the 1974 original.
Knapp W., Schmid P.: Codes with prescribed permutation group. J. Algebra 67(2), 415–435 (1980).
Li C.H.: The finite primitive permutation groups containing an abelian regular subgroup. Proc. Lond. Math. Soc. 87(3), 725–747 (2003).
Meldrum J.D.P.: Wreath Products of Groups and Semigroups, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 74. Longman, Harlow (1995).
Muzychuk M.: A solution of the isomorphism problem for circulant graphs. Proc. Lond. Math. Soc. 88(3), 1–41 (2004).
Muzychuk M.: A solution of an equivalence problem for semisimple cyclic codes, ArXiv:1105.4320v1 (2011).
Muzychuk M.: Ádám’s conjecture is true in the square-free case. J. Comb. Theory Ser. A 72(1), 118–134 (1995).
Muzychuk M.: On Ádám’s conjecture for circulant graphs. Discret. Math. 176(1–3), 285–298 (1997).
Muzychuk M.: On the isomorphism problem for cyclic combinatorial objects. Discret. Math. 197/198, 589–606 (1999). 16th British Combinatorial Conference (London, 1997).
Pálfy P.P.: Isomorphism problem for relational structures with a cyclic automorphism. Eur. J. Combin. 8(1), 35–43 (1987).
Sendrier N., Simos D.E.: How easy is code equivalence over \({\mathbb{F}}_{q}\)?, preprint.
Turner J.: Point-symmetric graphs with a prime number of points. J. Comb. Theory 3, 136–145 (1967).
Wielandt H.: Finite Permutation Groups, Translated from the German by R. Bercov, Academic Press, New York (1964).
Acknowledgments
The author is indebted as usual to the anonymous referees whose careful reading resulted in suggestions which greatly improved the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. E. Praeger.
Rights and permissions
About this article
Cite this article
Dobson, E. Monomial isomorphisms of cyclic codes. Des. Codes Cryptogr. 76, 257–267 (2015). https://doi.org/10.1007/s10623-014-9945-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-014-9945-5