American Institute of Mathematical Sciences

Advanced
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  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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

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
[2]

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
[3]

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
[4]

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
[5]

Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409
[6]

Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005
[7]

Yongbo Xia, Tor Helleseth, Chunlei Li. Some new classes of cyclic codes with three or six weights. Advances in Mathematics of Communications, 2015, 9 (1) : 23-36. doi: 10.3934/amc.2015.9.23
[8]

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
[9]

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
[10]

Yiwei Liu, Zihui Liu. On some classes of codes with a few weights. Advances in Mathematics of Communications, 2018, 12 (2) : 415-428. doi: 10.3934/amc.2018025
[11]

Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[12]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177
[13]

Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
[14]

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
[15]

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
[16]

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
[17]

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
[18]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55
[19]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025
[20]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (40)
  • HTML views (118)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences