Skew constacyclic codes over a non-chain ring $${\mathbb {F}}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle $$
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Skew constacyclic codes over a non‑chain ring 𝔽q [u, v]∕⟨f (u), g(v), uv − vu⟩ Swati Bhardwaj1 · Madhu Raka1 Received: 1 October 2019 / Revised: 2 March 2020 / Accepted: 3 March 2020 © Springer-Verlag GmbH, DE 2020
Abstract Let f(u) and g(v) be two polynomials, not both linear, which split into distinct linear factors over 𝔽q . Let R = 𝔽q [u, v]∕⟨f (u), g(v), uv − vu⟩ be a finite commutative nonchain ring. In this paper, we study general skew cyclic codes and 𝜃t-skew constacyclic codes over the ring R where 𝜃t is an automorphism of R. Keywords Skew polynomial ring · Skew cyclic codes · Skew quasi-cyclic codes · Quasi-twisted codes · Gray map Mathematics Subject Classification 94B15 · 11T71
1 Introduction Cyclic codes over finite fields have been studied since 1960’s because of their algebraic structures as ideals in certain commutative rings. Interest in codes over finite rings increased substantially after a break-through work by Hammons et al. in 1994. In 2007, Boucher et al. [4] generalized the concept of cyclic code over a non-commutative ring, namely skew polynomial ring 𝔽q [x;𝜃] , where 𝔽q is a field with q elements and 𝜃 is an automorphism of 𝔽q . In the polynomial ring 𝔽q [x;𝜃] , addition is defined as the usual one of polynomials and the multiplication is defined by the rule axi ∗ bxj = a𝜃 i (b)xi+j for a, b ∈ 𝔽q . Boucher and Ulmer [5] constructed some 𝜃 -cyclic codes called skew cyclic codes with Hamming distance larger than that of previously known linear codes with the same parameters. Siap et al. [22] investigated structural properties of skew cyclic codes of arbitrary length. After the first phase of study on skew cyclic codes over fields, the focus of attention moved to skew cyclic codes over rings. Abualrub et al. [1] studied skew * Madhu Raka [email protected] Swati Bhardwaj [email protected] 1
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160014, India
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cyclic codes over 𝔽2 + v𝔽2 , where v2 = v and the automorphism 𝜃 was taken as 𝜃 ∶ v → v + 1 . Jin [15] studied skew cyclic codes over 𝔽p + v𝔽p , where v2 = 1 with the automorphism 𝜃 taken as 𝜃 ∶ a + bv → a − bv . In 2014, Gursoy et al. [12] determined generator polynomials and found idempotent generators of skew cyclic codes over 𝔽q + v𝔽q , where v2 = v , q = pe and the automorphism 𝜃 was taken as t t 𝜃t ∶ a + bv → ap + bp v , 0 ≤ t ≤ e − 1 . Shi et al. [20] studied 𝜃t-skew-cyclic codes over 𝔽q + v𝔽q + v2 𝔽q , where v3 = v . Later Shi et al. [21] extended these results to skew cyclic codes over 𝔽q + v𝔽q + ⋯ + vm−1 𝔽q , where vm = v and q ≡ 1(mod m − 1) . Gao et al. [7] studied skew constacyclic codes over 𝔽q + v𝔽q , where v2 = v . Kabore et al. [17] studied skew constacyclic codes over 𝔽q [v]∕⟨vq − v⟩. Recently people have started studying skew cyclic codes over finite commutative non-chain rings having 2 or more variables. Yao et al. [23] studied skew cyclic codes over 𝔽q + u𝔽q + v𝔽q + uv𝔽q , where u2 = u, v2 = v, uv = vu and q is a prime power. Ashra
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