August  2020, 14(3): 455-466. doi: 10.3934/amc.2020028

On polycyclic codes over a finite chain ring

1. 

Faculty of Science, University of Yaoundé 1, Yaoundé, Cameroon

2. 

Institute of Mathematics, University of Valladolid, Edificio LUCIA - Campus Miguel Delibes, Valladolid 47011, Castilla, Spain

3. 

Department of Mathematics, Western Illinois University, 1 University Circle, Macomb, IL61455, USA

* Corresponding author

Received  November 2018 Published  September 2019

Fund Project: The second author is supported by the Spanish MINECO under Grant MTM2015-65764-C3-1

Galois images of polycyclic codes over a finite chain ring $ S $ and their annihilator dual are investigated. The case when a polycyclic code is Galois-disjoint over the ring $ S, $ is characterized and, the trace codes and restrictions of free polycyclic codes over $ S $ are also determined giving an analogue of Delsarte's theorem relating the trace code and the annihilator dual code.

Citation: 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
References:
[1]

A. AlahmadiS. DoughertyA. Leroy and P. Solé, On the duality and the direction of polycyclic codes, Adv. Math. Commun., 10 (2016), 921-929.  doi: 10.3934/amc.2016049.  Google Scholar

[2]

T. Blackford, The Galois variance of constacyclic codes, Finite Fields Appl., 47 (2017), 286-308.  doi: 10.1016/j.ffa.2017.06.001.  Google Scholar

[3] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[4]

A. Fotue Tabue and C. Mouaha, Contraction of cyclic codes over finite chain rings, Discrete Mathematics, 341 (2018), 1722-1731.  doi: 10.1016/j.disc.2018.03.008.  Google Scholar

[5]

A. Fotue Tabue, E. Martínez-Moro and C. Mouaha, Galois correspondence on linear codes over finite chain rings, in Discrete Mathematics. Google Scholar

[6]

X. -Dong HouS. R. Lopez-Permouth and B. Parra-Avila, Rational power series, sequential codes and periodicity of sequences, J. Pure Appl. Algebra, 213 (2009), 1157-1169.  doi: 10.1016/j.jpaa.2008.11.011.  Google Scholar

[7]

T. Kasami, Optimum shortened cyclic codes for burst-error correction, IEEE Trans. Inform. Theory, 9 (1963), 105-109.  doi: 10.1109/tit.1963.1057825.  Google Scholar

[8]

S. R. López-PermouthH. ÖzadamF. Özbudak and S. Szabo, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields Appl., 19 (2013), 16-38.  doi: 10.1016/j.ffa.2012.10.002.  Google Scholar

[9]

S. R. López-PermouthB. R. Parra-Avila and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun., 3 (2009), 227-234.  doi: 10.3934/amc.2009.3.227.  Google Scholar

[10]

B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 28 (1974).  Google Scholar

[11]

E. Martínez-MoroA. Piñera-Nicolás and F. I. Rúa, Codes over affine algebras with a finite commutative chain coefficient ring, Finite Fields Appl., 49 (2018), 94-107.  doi: 10.1016/j.ffa.2017.09.008.  Google Scholar

[12]

E. Martínez-MoroA. P. Nicolás and I. F. Rúa, On trace codes and Galois invariance over finite commutative chain rings, Finite Fields Appl., 22 (2013), 114-121.  doi: 10.1016/j.ffa.2013.03.004.  Google Scholar

[13]

A. A. Nechaev, Finite rings with applications, in: Handbook of Algebra, Handb. Algebr., Elsevier/North-Holland, Amsterdam, 5 (2008), 213–320. doi: 10.1016/S1570-7954(07)05005-X.  Google Scholar

[14]

G. H. Norton and A. Sǎlǎgean, On the structure of linear and cyclic codes over finite chain rings, Appl Algebra Eng Commun Comput., 10 (2000), 489-506.  doi: 10.1007/PL00012382.  Google Scholar

[15]

G. H. Norton and A. Sǎlǎgean, Cyclic codes and minimal strong Gröbner bases over a principal ideal ring, Finite Fields Appl., 9 (2003), 237-249.  doi: 10.1016/S1071-5797(03)00003-0.  Google Scholar

[16]

W. W. Peterson and E. J. Weldon, Error correcting codes, MIT Press, Cambridge, Mass.-London, 1972.  Google Scholar

[17]

Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/5350.  Google Scholar

show all references

References:
[1]

A. AlahmadiS. DoughertyA. Leroy and P. Solé, On the duality and the direction of polycyclic codes, Adv. Math. Commun., 10 (2016), 921-929.  doi: 10.3934/amc.2016049.  Google Scholar

[2]

T. Blackford, The Galois variance of constacyclic codes, Finite Fields Appl., 47 (2017), 286-308.  doi: 10.1016/j.ffa.2017.06.001.  Google Scholar

[3] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[4]

A. Fotue Tabue and C. Mouaha, Contraction of cyclic codes over finite chain rings, Discrete Mathematics, 341 (2018), 1722-1731.  doi: 10.1016/j.disc.2018.03.008.  Google Scholar

[5]

A. Fotue Tabue, E. Martínez-Moro and C. Mouaha, Galois correspondence on linear codes over finite chain rings, in Discrete Mathematics. Google Scholar

[6]

X. -Dong HouS. R. Lopez-Permouth and B. Parra-Avila, Rational power series, sequential codes and periodicity of sequences, J. Pure Appl. Algebra, 213 (2009), 1157-1169.  doi: 10.1016/j.jpaa.2008.11.011.  Google Scholar

[7]

T. Kasami, Optimum shortened cyclic codes for burst-error correction, IEEE Trans. Inform. Theory, 9 (1963), 105-109.  doi: 10.1109/tit.1963.1057825.  Google Scholar

[8]

S. R. López-PermouthH. ÖzadamF. Özbudak and S. Szabo, Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes, Finite Fields Appl., 19 (2013), 16-38.  doi: 10.1016/j.ffa.2012.10.002.  Google Scholar

[9]

S. R. López-PermouthB. R. Parra-Avila and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun., 3 (2009), 227-234.  doi: 10.3934/amc.2009.3.227.  Google Scholar

[10]

B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 28 (1974).  Google Scholar

[11]

E. Martínez-MoroA. Piñera-Nicolás and F. I. Rúa, Codes over affine algebras with a finite commutative chain coefficient ring, Finite Fields Appl., 49 (2018), 94-107.  doi: 10.1016/j.ffa.2017.09.008.  Google Scholar

[12]

E. Martínez-MoroA. P. Nicolás and I. F. Rúa, On trace codes and Galois invariance over finite commutative chain rings, Finite Fields Appl., 22 (2013), 114-121.  doi: 10.1016/j.ffa.2013.03.004.  Google Scholar

[13]

A. A. Nechaev, Finite rings with applications, in: Handbook of Algebra, Handb. Algebr., Elsevier/North-Holland, Amsterdam, 5 (2008), 213–320. doi: 10.1016/S1570-7954(07)05005-X.  Google Scholar

[14]

G. H. Norton and A. Sǎlǎgean, On the structure of linear and cyclic codes over finite chain rings, Appl Algebra Eng Commun Comput., 10 (2000), 489-506.  doi: 10.1007/PL00012382.  Google Scholar

[15]

G. H. Norton and A. Sǎlǎgean, Cyclic codes and minimal strong Gröbner bases over a principal ideal ring, Finite Fields Appl., 9 (2003), 237-249.  doi: 10.1016/S1071-5797(03)00003-0.  Google Scholar

[16]

W. W. Peterson and E. J. Weldon, Error correcting codes, MIT Press, Cambridge, Mass.-London, 1972.  Google Scholar

[17]

Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/5350.  Google Scholar

Figure 1.  Cyclicity of codes
[1]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

[2]

Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169

[3]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

[4]

Liqin Qian, Xiwang Cao. Character sums over a non-chain ring and their applications. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020134

[5]

Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021024

[6]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265

[7]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[8]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

[9]

Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269

[10]

Benrong Zheng, Xianpei Hong. Effects of take-back legislation on pricing and coordination in a closed-loop supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021035

[11]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (361)
  • HTML views (582)
  • Cited by (1)

[Back to Top]