American Institute of Mathematical Sciences

Advanced
August  2008, 2(3): 273-292. doi: 10.3934/amc.2008.2.273

Skew constacyclic codes over Galois rings

Delphine Boucher 1, , Patrick Solé 2, and  Felix Ulmer 1,

1. 

IRMAR (UMR 6625), Université de Rennes 1, Campus de Beaulieu, F-35042 Rennes, France, France
2. 

I3S, 2000 route des Lucioles, F-06903 Sophia Antipolis, France

Received  January 2008 Revised  July 2008 Published  July 2008

We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left and right Euclidean. Codes that are principal ideals in quotient rings of skew polynomial rings by a two sided ideals are studied. As an application, skew constacyclic self-dual codes over $GR(4, 2)$ are constructed. Euclidean self-dual codes give self-dual $\mathbb Z_4$−codes. Hermitian self-dual codes yield 3−modular lattices and quasi-cyclic self-dual $\mathbb Z_4$−codes.
Keywords: Cyclic codes, self-dual codes, skew polynomial rings, modular lattices., $\ZZ_4-$codes.
Mathematics Subject Classification: 94B60, 12H1.
Citation: 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
[1]

Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65
[2]

Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
[3]

Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[4]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[5]

Steven T. Dougherty, Joe Gildea, Adrian Korban, Abidin Kaya. Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68. Advances in Mathematics of Communications, 2019  doi: 10.3934/amc.2020037
[6]

Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035
[7]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229
[8]

Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267
[9]

Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047
[10]

Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020077
[11]

Delphine Boucher. Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$. Advances in Mathematics of Communications, 2016, 10 (4) : 765-795. doi: 10.3934/amc.2016040
[12]

Christos Koukouvinos, Dimitris E. Simos. Construction of new self-dual codes over $GF(5)$ using skew-Hadamard matrices. Advances in Mathematics of Communications, 2009, 3 (3) : 251-263. doi: 10.3934/amc.2009.3.251
[13]

Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039
[14]

Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2020, 14 (2) : 319-332. doi: 10.3934/amc.2020023
[15]

Somphong Jitman, San Ling, Ekkasit Sangwisut. On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$. Advances in Mathematics of Communications, 2016, 10 (2) : 255-273. doi: 10.3934/amc.2016004
[16]

Joaquim Borges, Steven T. Dougherty, Cristina Fernández-Córdoba. Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$. Advances in Mathematics of Communications, 2012, 6 (3) : 287-303. doi: 10.3934/amc.2012.6.287
[17]

W. Cary Huffman. Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order. Advances in Mathematics of Communications, 2007, 1 (3) : 357-398. doi: 10.3934/amc.2007.1.357
[18]

Ken Saito. Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs. Advances in Mathematics of Communications, 2019, 13 (2) : 213-220. doi: 10.3934/amc.2019014
[19]

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
[20]

Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (28)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences