On polycyclic codes over a finite chain ring

Alexandre Fotue-Tabue; Edgar Martínez-Moro; J. Thomas Blackford

doi:10.3934/amc.2020028

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2020, Volume 14, Issue 3: 455-466. Doi: 10.3934/amc.2020028

This issue Previous Article A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes Next Article A shape-gain approach for vector quantization based on flat tori

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
^* Corresponding author

Received: October 31, 2018

Early access: September 2019

Published: August 2020

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 13B02; Secondary: 94B05.

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Figure 1. Cyclicity of codes

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[1]	A. Alahmadi, S. Dougherty, A. 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.
[2]	T. Blackford, The Galois variance of constacyclic codes, Finite Fields Appl., 47 (2017), 286-308. doi: 10.1016/j.ffa.2017.06.001.
[3]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.
[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.
[5]	A. Fotue Tabue, E. Martínez-Moro and C. Mouaha, Galois correspondence on linear codes over finite chain rings, in Discrete Mathematics.
[6]	X. -Dong Hou, S. 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.
[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.
[8]	S. R. López-Permouth, H. Özadam, F. 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.
[9]	S. R. López-Permouth, B. 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.
[10]	B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 28 (1974).
[11]	E. Martínez-Moro, A. 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.
[12]	E. Martínez-Moro, A. 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.
[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.
[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.
[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.
[16]	W. W. Peterson and E. J. Weldon, Error correcting codes, MIT Press, Cambridge, Mass.-London, 1972.
[17]	Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/5350.