American Institute of Mathematical Sciences

Advanced
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

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.  doi: 10.1007/s10623-012-9696-0.  Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1504/IJICoT.2010.032133.  Google Scholar

[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., (1993).   Google Scholar

[6]

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

[7]

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

[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, (2012), 2904.  doi: 10.1109/ISIT.2012.6284057.  Google Scholar

[9]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511807077.  Google Scholar

[10]

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

[11]

S. R. López-Permouth and S. Szabo, Repeated root cyclic and negacyclic codes over Galois rings,, in Appl. Alg. Eng. Com. Comp., (2009), 219.  doi: 10.3934/amc.2009.3.409.  Google Scholar

[12]

F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups,, Ph.D. thesis, (1962).   Google Scholar

[13]

B. R. McDonald, Finite Rings with Identity,, Marcel Dekker, (1974).   Google Scholar

[14]

A. A. Nechaev and T. Khonol'd, Weighted modules and representations of codes (in Russian),, Probl. Peredachi Inform., 35 (1999), 18.   Google Scholar

[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.  doi: 10.1007/PL00012382.  Google Scholar

[16]

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

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.  doi: 10.1007/s10623-012-9696-0.  Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1504/IJICoT.2010.032133.  Google Scholar

[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., (1993).   Google Scholar

[6]

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

[7]

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

[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, (2012), 2904.  doi: 10.1109/ISIT.2012.6284057.  Google Scholar

[9]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511807077.  Google Scholar

[10]

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

[11]

S. R. López-Permouth and S. Szabo, Repeated root cyclic and negacyclic codes over Galois rings,, in Appl. Alg. Eng. Com. Comp., (2009), 219.  doi: 10.3934/amc.2009.3.409.  Google Scholar

[12]

F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups,, Ph.D. thesis, (1962).   Google Scholar

[13]

B. R. McDonald, Finite Rings with Identity,, Marcel Dekker, (1974).   Google Scholar

[14]

A. A. Nechaev and T. Khonol'd, Weighted modules and representations of codes (in Russian),, Probl. Peredachi Inform., 35 (1999), 18.   Google Scholar

[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.  doi: 10.1007/PL00012382.  Google Scholar

[16]

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

[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]

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
[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]

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
[5]

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
[6]

Anderson Silva, C. Polcino Milies. Cyclic codes of length $ 2p^n $ over finite chain rings. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020017
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

Alexandre Fotue-Tabue, Edgar Martínez-Moro, J. Thomas Blackford. On polycyclic codes over a finite chain ring. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020028
[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]

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
[15]

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
[16]

Ekkasit Sangwisut, Somphong Jitman, Patanee Udomkavanich. Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields. Advances in Mathematics of Communications, 2017, 11 (3) : 595-613. doi: 10.3934/amc.2017045
[17]

Somphong Jitman, Ekkasit Sangwisut. The average dimension of the Hermitian hull of constacyclic codes over finite fields of square order. Advances in Mathematics of Communications, 2018, 12 (3) : 451-463. doi: 10.3934/amc.2018027
[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]

Anuradha Sharma, Saroj Rani. Trace description and Hamming weights of irreducible constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 123-141. doi: 10.3934/amc.2018008
[20]

Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (38)
  • HTML views (1)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences