Introduction to Coding Theory
A Fast Matrix Decoding Algorithm for Rank-Error-Correcting Codes
Proceedings of the First French-Soviet Workshop on Algebraic Coding
Applicable Algebra in Engineering, Communication and Computing
Skew codes of prescribed distance or rank
Designs, Codes and Cryptography
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
A welch–berlekamp like algorithm for decoding gabidulin codes
WCC'05 Proceedings of the 2005 international conference on Coding and Cryptography
Isometries for rank distance and permutation group of Gabidulin codes
IEEE Transactions on Information Theory
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In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several approaches to generalize Reed-Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed-Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank metric and derive families of Maximum Distance Separable and Maximum Rank Distance codes.