Some constacyclic codes over finite chain rings

Previous Article
Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights
AMC Home
This Issue
Next Article

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

Some constacyclic codes over finite chain rings

Aicha Batoul ^1, , Kenza Guenda ^2, and T. Aaron Gulliver ^3,

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

Abstract
Related Papers

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 $GR(p^e,r)$.

Keywords: equivalent codes., Finite chain rings, constacyclic codes.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

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.

[1]	Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39
[2]	Gianira N. Alfarano, Anina Gruica, Julia Lieb, Joachim Rosenthal. Convolutional codes over finite chain rings, MDP codes and their characterization. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022028
[3]	Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273
[4]	Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Esengül Saltürk. Skew constacyclic codes over the local Frobenius non-chain rings of order 16. Advances in Mathematics of Communications, 2020, 14 (1) : 53-67. doi: 10.3934/amc.2020005
[5]	Hai Q. Dinh, Hien D. T. Nguyen. On some classes of constacyclic codes over polynomial residue rings. Advances in Mathematics of Communications, 2012, 6 (2) : 175-191. doi: 10.3934/amc.2012.6.175
[6]	Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2020, 14 (2) : 233-245. doi: 10.3934/amc.2020017
[7]	Ferruh Özbudak, Patrick Solé. Gilbert-Varshamov type bounds for linear codes over finite chain rings. Advances in Mathematics of Communications, 2007, 1 (1) : 99-109. doi: 10.3934/amc.2007.1.99
[8]	Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395
[9]	Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543
[10]	Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039
[11]	Zihui Liu, Dajian Liao. Higher weights and near-MDR codes over chain rings. Advances in Mathematics of Communications, 2018, 12 (4) : 761-772. doi: 10.3934/amc.2018045
[12]	Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039
[13]	Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035
[14]	Alexandre Fotue-Tabue, Edgar Martínez-Moro, J. Thomas Blackford. On polycyclic codes over a finite chain ring. Advances in Mathematics of Communications, 2020, 14 (3) : 455-466. doi: 10.3934/amc.2020028
[15]	David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131
[16]	Ram Krishna Verma, Om Prakash, Ashutosh Singh, Habibul Islam. New quantum codes from skew constacyclic codes. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021028
[17]	Ranya Djihad Boulanouar, Aicha Batoul, Delphine Boucher. An overview on skew constacyclic codes and their subclass of LCD codes. Advances in Mathematics of Communications, 2021, 15 (4) : 611-632. doi: 10.3934/amc.2020085
[18]	Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018
[19]	Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004
[20]	Zihui Liu. Galois LCD codes over rings. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022002

2021 Impact Factor: 1.015

Tools

Metrics

Other articles
by authors

on AIMS
Aicha Batoul Kenza Guenda T. Aaron Gulliver
on Google Scholar
Aicha Batoul Kenza Guenda T. Aaron Gulliver

[Back to Top]