Abstract
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.
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Communicated by J. Jedwab.
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Chaussade, L., Loidreau, P. & Ulmer, F. Skew codes of prescribed distance or rank. Des. Codes Cryptogr. 50, 267–284 (2009). https://doi.org/10.1007/s10623-008-9230-6
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DOI: https://doi.org/10.1007/s10623-008-9230-6