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Skew constacyclic codes over finite chain rings

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  • Skew polynomial rings over finite fields and over Galois rings have recently been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is a unit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Specializing to codes over the ring $\mathbb F$pm$+u\mathbb F$pm, the structure of all skew constacyclic codes is completely determined. This allows us to express the generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length $2$ over $\mathbb F_3 + u\mathbb F_3$ and their Euclidean and Hermitian dual codes is also provided.
    Mathematics Subject Classification: Primary: 94B15, 13B25; Secondary: 94B60.

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