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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
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