10,342 research outputs found

    Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring

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    Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed-Solomon codes, has already been introduced in the literature. In this work, we introduce a linearized version of such codes, called linearized Reed-Solomon codes. We prove that they have maximum sum-rank distance. Such distance is of interest in multishot network coding or in singleshot multi-network coding. To prove our result, we introduce new metrics defined by skew polynomials, which we call skew metrics, we prove that skew Reed-Solomon codes have maximum skew distance, and then we translate this scenario to linearized Reed-Solomon codes and the sum-rank metric. The theories of Reed-Solomon codes and Gabidulin codes are particular cases of our theory, and the sum-rank metric extends both the Hamming and rank metrics. We develop our theory over any division ring (commutative or non-commutative field). We also consider non-zero derivations, which give new maximum rank distance codes over infinite fields not considered before

    Skew Cyclic codes over \F_q+u\F_q+v\F_q+uv\F_q

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    In this paper, we study skew cyclic codes over the ring R=\F_q+u\F_q+v\F_q+uv\F_q, where u2=u,v2=v,uv=vuu^{2}=u,v^{2}=v,uv=vu, q=pmq=p^{m} and pp is an odd prime. We investigate the structural properties of skew cyclic codes over RR through a decomposition theorem. Furthermore, we give a formula for the number of skew cyclic codes of length nn over $R.
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