Abstract
We obtain structural results about group ring codes over F[G], where F is a finite field of characteristic p > 0 and the Sylow p-subgroup of the Abelian group G is cyclic. As a special case, we characterize cyclic codes over finite fields in the case the length of the code is divisible by the characteristic of the field. By the same approach we study cyclic codes of length m over the ring R = F q [u], u r = 0 with r > 0, gcd(m, q) = 1. Finally, we give a construction of quasi-cyclic codes over finite fields.
Similar content being viewed by others
References
Abualrub T., Siap I.: Cyclic codes over the rings Z 2 + uZ 2 and Z 2 + uZ 2 + u 2 Z 2. Des. Codes Cryptogr. 42, 273–287 (2007)
Berman S.D.: On the theory of group codes. Kibernetika 3, 31–39 (1967)
Berman S.D.: Semi-simple cyclic and Abelian codes. Kibernetika 3, 21–30 (1967)
Bonnecaze A., Udaya P.: Cyclic codes and self-dual codes over F 2 + uF 2. IEEE Trans. Inform. Theory 45, 1250–1255 (1999)
Camion P.: Abelian codes. University of Wisconsin, Madison, Mathematics Research Center Technical Report (1971).
Castagnoli G., Massey J.L., Schoeller P., von Seemann N.: On repeated-root cyclic codes. IEEE Trans. Inform. Theory 37, 337–342 (1991)
Chabanne H.: Permutation decoding of Abelian Codes. IEEE Trans. Inform. Theory 38, 1826–1829 (1992)
Conan J., Séguin G.E.: Structural properties and enumeration of quasi-cyclic codes. AAECC 4, 25–39 (1993)
Delsarte P.: Automorphisms of Abelian Codes. Philips Res. Rep. 25, 389–402 (1970)
Feng T.: Relative (pn, p, pn, n)-difference sets with gcd(p, n) = 1. J. Algebraic Combin., to appear. http://www.springerlink.com/content/47727702334975n7/fulltext.pdf.
Lally K., Fitzpatrick P.: Algebraic structure of quasi-cyclic codes. Discrete Appl. Math. 111, 157–175 (1970)
Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes I: finite fields. IEEE Trans. Inform. Theory 47, 2751–2760 (2001)
Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes II: chain rings. Des. Codes Cryptogr. 30, 113–130 (2003)
Ling S., : On the algebraic structure of quasi-cyclic codes III: generator theory. IEEE Trans. Inform. Theory 51, 2652–2700 (2005)
Ling S., Solé P.: On the algebraic structure of quasi-cyclic codes IV: repeated roots. Des. Codes Cryptogr. 38, 337–361 (2006)
Macwilliams F.J.: Binary codes which are ideals in the group algebra of an Abelian group. Bell Syst. Tech. J. 49, 987–1011 (1970)
Macwilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
Passman D.S.: The Algebraic Structure of Group Rings. Wiley, New York (1977)
Rajan B.S., Siddiqi M.U.: Transform domain characterization of Abelian codes. IEEE Trans. Inform. Theory 38, 1817–1821 (1992)
Séguin G.E.: A class of 1-generator quasi-cyclic codes. IEEE Trans. Inform. Theory 50, 1745–1753 (2004)
Sloane N.J.A., Thompson J.G.: Cyclic self-dual codes. IEEE Trans. Inform. Theory 29, 364–366 (1983)
van Lint J.H.: Repeated-root cyclic codes. IEEE Trans. Inform. Theory 37, 343–345 (1991)
Willems W.: A note on self-dual group codes. IEEE Trans. Inform. Theory 48, 3107–3109 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. McGuire.
Rights and permissions
About this article
Cite this article
Fu, W., Feng, T. On self-orthogonal group ring codes. Des. Codes Cryptogr. 50, 203–214 (2009). https://doi.org/10.1007/s10623-008-9224-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-008-9224-4