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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, 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
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