November  2016, 10(4): 683-694. doi: 10.3934/amc.2016034

Some constacyclic codes over finite chain rings

1. 

Faculty of Mathematics, USTHB, Algiers, Algeria

2. 

Faculty of Mathematics, University of Science and Technology, USTHB, Algeria

3. 

Department of Electrical and Computer Engineering, University of Victoria, PO Box 1700, STN CSC, Victoria, BC, Canada

Received  February 2014 Revised  December 2014 Published  November 2016

We give the structure of constacyclic codes over some chain rings. We also provide conditions on the equivalence between constacyclic codes and cyclic codes over finite chain rings.As a special case,we consider the structure of $(\alpha + \beta p)$-constacyclic codes of length $p^s$ over $GR(p^e,r)$.
Citation: Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034
References:
[1]

A. Batoul, K. Guenda and T. A. Gulliver, On self-dual cyclic codes over finite chain rings, Des. Codes Cryptogr., 70 (2014), 347-358. doi: 10.1007/s10623-012-9696-0.

[2]

H. Dinh, On the linear ordering of some classes of negacyclic and cyclic codes and their distributions, Finite Fields Appl., 14 (2008), 22-40. doi: 10.1016/j.ffa.2007.07.001.

[3]

H. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744. doi: 10.1109/TIT.2004.831789.

[4]

S. T. Dougherty, J. L. Kim and H. Liu, Construction of self-dual codes over finite commutative chain rings, Int. J. Inform. Coding Theory, 1 (2010), 171-190. doi: 10.1504/IJICoT.2010.032133.

[5]

G. D. Forney, N. J. A. Sloane and M. Trott, The Nordstrom-Robinson code is the binary image of the octacode, in DIMACS/IEEE Workshop Coding Quantiz., Amer. Math. Soc., 1993.

[6]

M. Greferath and S. E. Shmidt, Finite-ring combinatorics and Macwilliam's equivalence theorem, J. Combin. Theory A, 92 (2000), 17-28. doi: 10.1006/jcta.1999.3033.

[7]

K. Guenda and T. A. Gulliver, MDS and self-dual codes over rings, Finite Fields Appl., 18 (2012), 1061-1075. doi: 10.1016/j.ffa.2012.09.003.

[8]

K. Guenda and T. A. Gulliver, Self-dual repeated root cyclic and negacyclic codes over finite fields, in Proc. IEEE Int. Symp. Inform. Theory, Boston, 2012, 2904-2908. doi: 10.1109/ISIT.2012.6284057.

[9]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, New York, 2003. doi: 10.1017/CBO9780511807077.

[10]

P. Kanwar and S. R. López-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl., 3 (1997), 334-352. doi: 10.1006/ffta.1997.0189.

[11]

S. R. López-Permouth and S. Szabo, Repeated root cyclic and negacyclic codes over Galois rings, in Appl. Alg. Eng. Com. Comp., Springer-Verlag, New York, 2009, 219-222. doi: 10.3934/amc.2009.3.409.

[12]

F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups, Ph.D. thesis, Radcliffe College, Cambridge, MA, 1962.

[13]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974.

[14]

A. A. Nechaev and T. Khonol'd, Weighted modules and representations of codes (in Russian), Probl. Peredachi Inform., 35 (1999), 18-39; translation in Problems Inform. Transm., 35 (1999), 205-223.

[15]

G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Engr. Comm. Comput., 10 (2000), 489-506. doi: 10.1007/PL00012382.

[16]

J. Wood, Extension theorems for linear codes over finite rings, in Appl. Alg. Eng. Com. Comp. (eds. T. Mora and H. Matson), Springer-Verlag, New York, 1997, 329-340. doi: 10.1007/3-540-63163-1_26.

show all references

References:
[1]

A. Batoul, K. Guenda and T. A. Gulliver, On self-dual cyclic codes over finite chain rings, Des. Codes Cryptogr., 70 (2014), 347-358. doi: 10.1007/s10623-012-9696-0.

[2]

H. Dinh, On the linear ordering of some classes of negacyclic and cyclic codes and their distributions, Finite Fields Appl., 14 (2008), 22-40. doi: 10.1016/j.ffa.2007.07.001.

[3]

H. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744. doi: 10.1109/TIT.2004.831789.

[4]

S. T. Dougherty, J. L. Kim and H. Liu, Construction of self-dual codes over finite commutative chain rings, Int. J. Inform. Coding Theory, 1 (2010), 171-190. doi: 10.1504/IJICoT.2010.032133.

[5]

G. D. Forney, N. J. A. Sloane and M. Trott, The Nordstrom-Robinson code is the binary image of the octacode, in DIMACS/IEEE Workshop Coding Quantiz., Amer. Math. Soc., 1993.

[6]

M. Greferath and S. E. Shmidt, Finite-ring combinatorics and Macwilliam's equivalence theorem, J. Combin. Theory A, 92 (2000), 17-28. doi: 10.1006/jcta.1999.3033.

[7]

K. Guenda and T. A. Gulliver, MDS and self-dual codes over rings, Finite Fields Appl., 18 (2012), 1061-1075. doi: 10.1016/j.ffa.2012.09.003.

[8]

K. Guenda and T. A. Gulliver, Self-dual repeated root cyclic and negacyclic codes over finite fields, in Proc. IEEE Int. Symp. Inform. Theory, Boston, 2012, 2904-2908. doi: 10.1109/ISIT.2012.6284057.

[9]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, New York, 2003. doi: 10.1017/CBO9780511807077.

[10]

P. Kanwar and S. R. López-Permouth, Cyclic codes over the integers modulo $p^m$, Finite Fields Appl., 3 (1997), 334-352. doi: 10.1006/ffta.1997.0189.

[11]

S. R. López-Permouth and S. Szabo, Repeated root cyclic and negacyclic codes over Galois rings, in Appl. Alg. Eng. Com. Comp., Springer-Verlag, New York, 2009, 219-222. doi: 10.3934/amc.2009.3.409.

[12]

F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups, Ph.D. thesis, Radcliffe College, Cambridge, MA, 1962.

[13]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974.

[14]

A. A. Nechaev and T. Khonol'd, Weighted modules and representations of codes (in Russian), Probl. Peredachi Inform., 35 (1999), 18-39; translation in Problems Inform. Transm., 35 (1999), 205-223.

[15]

G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Engr. Comm. Comput., 10 (2000), 489-506. doi: 10.1007/PL00012382.

[16]

J. Wood, Extension theorems for linear codes over finite rings, in Appl. Alg. Eng. Com. Comp. (eds. T. Mora and H. Matson), Springer-Verlag, New York, 1997, 329-340. doi: 10.1007/3-540-63163-1_26.

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