The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Applicable Algebra in Engineering, Communication and Computing
Coding with skew polynomial rings
Journal of Symbolic Computation
Codes as Modules over Skew Polynomial Rings
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
Skew quasi-cyclic codes over Galois rings
Designs, Codes and Cryptography
A note on the dual codes of module skew codes
IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
Linear codes using skew polynomials with automorphisms and derivations
Designs, Codes and Cryptography
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In this paper we generalize the notion of cyclic code and construct codes as ideals in finite quotients of non-commutative polynomial rings, so called skew polynomial rings of automorphism type. We propose a method to construct block codes of prescribed rank and a method to construct block codes of prescribed distance. Since there is no unique factorization in skew polynomial rings, there are much more ideals and therefore much more codes than in the commutative case. In particular we obtain a [40, 23, 10]4 code by imposing a distance and a [42,14,21]8 code by imposing a rank, which both improve by one the minimum distance of the previously best known linear codes of equal length and dimension over those fields. There is a strong connection with linear difference operators and with linearized polynomials (or q-polynomials) reviewed in the first section.