$\mathbb{Z}_{q}(\mathbb{Z}_{q}+u\mathbb{Z}_{q})-$ linear skew constacyclic codes

Year 2020, Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections), 85 - 101, 29.02.2020

Ahlem Melakhessou Nuh Aydin , Zineb Hebbache Kenza Guenda

https://doi.org/10.13069/jacodesmath.671815

Cited By: 6

Abstract

In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0.$ We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^{\alpha}R^{\beta}$. Some structural properties of the skew polynomial ring $ R[x,\Theta]$ are discussed, where $ \Theta$ is an automorphism of $R.$ We describe the generator polynomials of skew constacyclic codes over $\mathbb{Z}_{q}R,$ also we determine their minimal spanning sets and their sizes. Further, by using the Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_{4}$. Finally, we have generalized these codes to double skew constacyclic codes over $\mathbb{Z}_{q}R$.

Keywords

Linear codes, Skew constacyclic codes, $\mathbb{Z}_{q}\mathbb{Z}_{q}, Bounds

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