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May  2020, 14(2): 233-245. doi: 10.3934/amc.2020017

Cyclic codes of length $ 2p^n $ over finite chain rings

Anderson Silva 1, and  C. Polcino Milies 2,

1. 

Departamento de Matemática, Universidade Federal de Viçosa, Viçosa, 36570-000, Brazil
2. 

Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 05311-970, Brazil

Received  February 2018 Revised  March 2019 Published  May 2020 Early access  September 2019

Fund Project: This work was partially supported by CNPq., Proc. 300243/79-0(RN) and FAPESP, Proc 2015/09162-9

We use group algebra methods to study cyclic codes over finite chain rings and under some restrictive hypotheses, described in section 2, for codes of length $ 2p^n $, $ p $ a prime, we are able to compute the minimum weights of all possible cyclic codes of that length.

Keywords: Cyclic codes, chain rings, weights.
Mathematics Subject Classification: Primary: 94B15, 06F25, 20C05; Secondary: 94A05, 94B05.
Citation: 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
References:
[1]

S. K. Arora and M. Pruthi, Minimal cyclic codes of length $2p^n$, Finite Fields Appl., 5 (1999), 177-187.  doi: 10.1006/ffta.1998.0238.

[2]

Y. L. Cao, On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.  doi: 10.1016/j.ffa.2013.07.001.

[3]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Transactions on Information Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.

[4]

S. T. Dougherty, J.-L. Kim and H. W. Liu, Construction of self-dual codes over finite commutative chain rings, Int. Journal on Information and Coding Theory, 1 (2010), 171-190.  doi: 10.1504/IJICOT.2010.032133.

[5]

R. A. Ferraz and C. Polcino Milies, Idempotents in group algebras and minimal abelian codes, Finite Fields and Their Appl., 13 (2007), 382-393.  doi: 10.1016/j.ffa.2005.09.007.

[6]

N. Jacobson, Basic Algebra. II, W. H. Freeman and Company, San Francisco, Calif., 1980.

[7]

Z. H. Liu, Notes on linear codes over finite chain rings, Acta Mathematicae Applicatae Sinica, 27 (2011), 141-148.  doi: 10.1007/s10255-011-0047-0.

[8]

E. Martinez-Moro and I. F. Rúa, On repeated-root multivariable codes over a finite chain ring, Designs, Codes Cryptography, 45 (2007), 219-227.  doi: 10.1007/s10623-007-9114-1.

[9]

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

[10]

C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Algebra and Applications, 1. Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-010-0405-3.

[11]

P. Solé and V. Sison, Bounds on the minimum homogeneous dis-tance of the $p^r$-ary image of linear block codes over the galois ring $GR(p^r, m)$, IEEE Trans. Information Theory, 53 (2007), 2270-2273.  doi: 10.1109/TIT.2007.896891.

show all references

References:
[1]

S. K. Arora and M. Pruthi, Minimal cyclic codes of length $2p^n$, Finite Fields Appl., 5 (1999), 177-187.  doi: 10.1006/ffta.1998.0238.

[2]

Y. L. Cao, On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.  doi: 10.1016/j.ffa.2013.07.001.

[3]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Transactions on Information Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.

[4]

S. T. Dougherty, J.-L. Kim and H. W. Liu, Construction of self-dual codes over finite commutative chain rings, Int. Journal on Information and Coding Theory, 1 (2010), 171-190.  doi: 10.1504/IJICOT.2010.032133.

[5]

R. A. Ferraz and C. Polcino Milies, Idempotents in group algebras and minimal abelian codes, Finite Fields and Their Appl., 13 (2007), 382-393.  doi: 10.1016/j.ffa.2005.09.007.

[6]

N. Jacobson, Basic Algebra. II, W. H. Freeman and Company, San Francisco, Calif., 1980.

[7]

Z. H. Liu, Notes on linear codes over finite chain rings, Acta Mathematicae Applicatae Sinica, 27 (2011), 141-148.  doi: 10.1007/s10255-011-0047-0.

[8]

E. Martinez-Moro and I. F. Rúa, On repeated-root multivariable codes over a finite chain ring, Designs, Codes Cryptography, 45 (2007), 219-227.  doi: 10.1007/s10623-007-9114-1.

[9]

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

[10]

C. Polcino Milies and S. K. Sehgal, An Introduction to Group Rings, Algebra and Applications, 1. Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-010-0405-3.

[11]

P. Solé and V. Sison, Bounds on the minimum homogeneous dis-tance of the $p^r$-ary image of linear block codes over the galois ring $GR(p^r, m)$, IEEE Trans. Information Theory, 53 (2007), 2270-2273.  doi: 10.1109/TIT.2007.896891.

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