A class of constacyclic codes and skew constacyclic codes over $$\pmb {\mathbb {Z}}_{2^s}+u\pmb {\mathbb {Z}}_{2^s}$$ and their gray images

Abstract

In this paper, we study (1+2 s−1 u)-constacyclic codes and a class of skew (1+2 s−1 u)constacyclic codes of odd length over the ring R = Z 2 s + uZ 2 s , u 2 = 0, where s ≥ 3 is an odd integer. We have obtained the algebraic structure of (1+2 s−1 u)-constacyclic codes over R. Three new Gray maps from R to Z 2 + uZ 2 have been defined and it is shown that Gray images of (1 + 2 s−1 u)-constacyclic codes and skew (1 + 2 s−1 u)constacyclic codes are cyclic codes, quasi-cyclic codes or codes that are permutation equivalent to quasi-cyclic codes over Z 2 + uZ 2 . Using Magma, some good cyclic codes of length 6 over Z 2 + uZ 2 are obtained.

Quantum codes from constacyclic codes over $S_{k}$

“…Let C be a (2u 2 1 -1)-constacyclic code of length 3 over S 1 with generator polynomial e 1 g 1 (x) + e 2 g 2 (x) + e 3 g 3 (x), where g 1 = x + 3, g 2 = x + 5, g 3 = x + 4. By Theorem 8, we have C ⊥ ⊆ C, and φ 1 (C) is a linear code over F 7 with parameters [9,6,2]. By Theorem 9, we know that there is a quantum error correcting code with parameters [[9, 3, ≥ 2]] 7 .…”

“…Calderbank et al [1] gave a way to construct quantum error correcting codes from classical error correcting codes, constructing quantum error correcting codes is a systematic and effective mathematical method by using constacyclic codes. There are a lot of works about constacyclic codes over finite fields and finite rings [2][3][4][5][6][7][8][9][10] and many good quantum codes constructed by using cyclic codes over finite rings [11][12][13][14]. Currently, some authors have obtained quantum codes from constacyclic codes over finite non-chain ring.…”