A class of constacyclic codes and skew constacyclic codes over $$\pmb {\mathbb {Z}}_{2^s}+u\pmb {\mathbb {Z}}_{2^s}$$
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A class of constacyclic codes and skew constacyclic codes Z2s and their gray images over Z 2s + uZ Raj Kumar1 · Maheshanand Bhaintwal1 Received: 6 April 2020 / Revised: 13 August 2020 / Accepted: 17 August 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we study (1+2s−1 u)-constacyclic codes and a class of skew (1+2s−1 u)constacyclic codes of odd length over the ring R = Z2s + uZ2s , u 2 = 0, where s ≥ 3 is an odd integer. We have obtained the algebraic structure of (1+2s−1 u)-constacyclic codes over R. Three new Gray maps from R to Z2 + uZ2 have been defined and it is shown that Gray images of (1 + 2s−1 u)-constacyclic codes and skew (1 + 2s−1 u)constacyclic codes are cyclic codes, quasi-cyclic codes or codes that are permutation equivalent to quasi-cyclic codes over Z2 + uZ2 . Using Magma, some good cyclic codes of length 6 over Z2 + uZ2 are obtained. Keywords Cyclic codes · Constacyclic codes · Quasi-cyclic codes · Skew constacyclic codes · Gray images Mathematics Subject Classification 94B05 · 94B15
1 Introduction Codes over finite rings have been studied in recent years [1,2,5,7,16,19,20,33]. Cyclic codes are important class of linear codes because of their rich algebraic structure, which makes this class easy to understand and implement. Cyclic codes over a finite ring have been extensively studied over the last few years, motivated by the paper of Hammons et al. [11], wherein it was shown that some well known binary non-linear codes are actually images of some linear codes over Z4 under the Gray map. Constacyclic codes are one of the important generalizations of cyclic codes, and it can be efficiently implemented by shift registers. In many cases, it can be observed that the constacyclic
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Raj Kumar [email protected] Maheshanand Bhaintwal [email protected]
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Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India
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R. Kumar, M. Bhaintwal
code is an excellent choice instead of a cyclic code to obtain some codes with better parameters. Constacyclic codes over finite fields were initiated by Berlekamp in the early 1960s [4], and have been studied extensively since then. Abualrub and Siap [3] studied constacyclic codes over the ring F2 +uF2 . Dinh [9] studied constacyclic codes of length 2s over Galois extension ring of F2 +uF2 , where s is a positive integer. Shi et al. [24] constructed two new infinite families of trace codes of dimension 2m over the ring F p + uF p , u 2 = u, where p is an odd prime. Many authors have also studied trace codes over finite rings [17,18,26–32]. Dinh [8] studied constacyclic codes of length p s over F pm + uF pm , where p is a prime and m and s are positive integers. Yildiz and Karadeniz introduced the ring R = F2 + uF2 + vF2 + uvF2 , where u 2 = v 2 = 0 and uv = vu [34–36], and derived the generator polynomials of cyclic codes over R and obtained some good binary codes as the images of these codes under two Gray maps. They have also studied (1 + v)-constacyclic codes of odd lengths
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