Abstract
We investigate negacyclic codes over the Galois ring GR(2a,m) of length N = 2k n, where n is odd and k ⩾ 0. We first determine the structure of u-constacyclic codes of length n over the finite chain ring \(GR(2^a ,m)[u]/\langle u^{2^k } + 1\rangle \). Then using a ring isomorphism we obtain the structure of negacyclic codes over GR(2a,m) of length N = 2k n (n odd) and explore the existence of self-dual negacyclic codes over GR(2a,m). A bound for the homogeneous distance of such negacyclic codes is also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berlekamp E R. Negacyclic codes for the Lee metric. In: Proc Conf Combinatorial Mathematics and Its Applications. Chapel Hill, NC, 1968: 298–316
Berlekamp E R. Algebraic Coding Theory. Laguna Hills, CA: Aegean Park Press, 1984
Blackford T. Negacyclic codes over ℤ4 of even length. IEEE Trans Inform Theory, 2003, 49: 1417–1424
Dinh H Q. Negacyclic codes of length 2s over Galois rings. IEEE Trans Inform Theory, 2005, 51: 4252–4262
Dinh H Q. Complete distances of all negacyclic codes of length 2s over \(\mathbb{Z}_{2^a } \). IEEE Trans Inform Theory, 2007, 53: 147–161
Dinh H Q, López-Permouth S R. Cyclic and negacyclic codes over finite chain rings. IEEE Trans Inform Theory, 2004, 50: 1728–1744
Dougherty S T, Kim J L, Liu H. Constructions of self-dual codes over finite commutative chain rings. Int J Inform Coding Theory, 2010, 1: 171–190.
Greferath M, Schmidt S E. Gray isometries for finite chain rings and a nonlinear ternary (36, 312, 15) code. IEEE Trans Inform Theory, 1999, 45: 2522–2524
Greferath M, Schmidt S E. Finite ring combinatorics and MacWilliams’s equivalence theorem. J Combin Theory Ser A, 2000, 92: 17–28
Greferath M, O’sullivan M E. On bounds for codes over Frobenius rings under homogeneous weights. Discr Math, 2004, 289: 11–24
Hammons Jr A R, Kumar P V, Calderbank A R, et al. The ℤ4-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans Inform Theory, 1994, 40: 301–319
Ling S, Solé P. On the algebraic structure of quasi-cyclic codes I: Finite field. IEEE Trans Inform Theory, 2001, 47: 2751–2760
McDonald B R. Finite Rings with Identity. New York: Marcel Dekker Press, 1974
Norton G H, Sãlãgean A. On the structure of linear and cyclic codes over a finite chain ring. Appl Algebra Eng Commun Comput, 2000, 10: 489–506
Sãlãgean A. Repeated-root cyclic and negacyclic codes over finite chain rings. Discr Appl Math, 2006, 154: 413–419
Solé P, Sison V. Bounds on the minimum homogeneous distance of the p r-ary image of linear block codes over the Galois ring GR(p r,m). IEEE Trans Inform Theory, 2007, 53: 2270–2273
Wan Z X. Quaternary Codes. Singapore: World Scientific Press, 1997
Wolfmann J. Negacyclic and cyclic codes over ℤ4. IEEE Trans Inform Theory, 1999, 45: 2527–2532
Wolfmann J. Binary images of cyclic codes over ℤ4. IEEE Trans Inform Theory, 2001, 47: 1773–1779
Wood J. Duality for modules over finite rings and applications to coding theory. Amer J Math, 1999, 121: 555–575
Zhu S X, Kai X S. Dual and self-dual negacyclic codes of even length over \(\mathbb{Z}_{2^a } \). Discr Math, 2009, 309: 2382–2391
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, S., Kai, X. Negacyclic codes over Galois rings of characteristic 2a . Sci. China Math. 55, 869–879 (2012). https://doi.org/10.1007/s11425-011-4309-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-011-4309-3