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Chinese Journal of Electronics

A Family of Constacyclic Codes over 𝔽 2 m + u 𝔽 2 m and Its Application to Quantum Codes

Yongsheng Tang, Ting Yao, Shixin Zhu, Xiaoshan Kai

Published Jan 1, 2020• 4 citations
Publisher

Abstract

Let R be the ring 𝔽2m + u𝔽2m, where u2 = 0. We introduce a Gray map from R to 𝔽22m and study (1 + u)-constacyclic codes over R. It is proved that the image of a (1 + u)-constacyclic code length n over R under the Gray map is a distance-invariant binary quasicyclic code of index m and length 2mn. We also prove that every code of length 2mn which is the Gray image of cyclic codes over R of length n is permutation equivalent to a binary quasi-cyclic code of index m. Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to (1 + u)-constacyclic codes over R.

DOI
10.1049/cje.2019.10.007

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Let R be the ring 𝔽2m + u𝔽2m, where u2 = 0. We introduce a Gray map from R to 𝔽22m and study (1 + u)-constacyclic codes over R. It is proved that the image of a (1 + u)-constacyclic code length n over R under the Gray map is a distance-invariant binary quasicyclic code of index m and length 2mn. We also prove that every code of length 2mn which is the Gray image of cyclic codes over R of length n is permutation equivalent to a binary quasi-cyclic code of index m. Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to (1 + u)-constacyclic codes over R.
A family of constacyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ and application to quantum codes
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A family of constacyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}F2m​+uF2m​ and its application to quantum codes
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In this paper, we study the structural properties of ( α + u 1 β + u 2 γ + u 1 u 2 δ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / ⟨ u 1 2 − u 1 , u 2 2 − u 2 , u 1 u 2 − u 2 u 1 ⟩ where q = p m for odd prime p and m ≥ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .
We study the algebraic structures of constacyclic codes over a new finite non-chain ring R and their l-Galois dual codes based on a new Gray map ϕ and determine the dimensions of the l-Galois hull of constacyclic codes over R. Furthermore, we propose a quantum construction X of the Galois inner product to construct quantum codes by combining the Calderbank–Shor–Steane (CSS) construction, Hermitian construction and construction X. As an application, by calculating the dimensions of the l-Galois hull of constacyclic codes over R, we obtain some new quantum maximum distance separable (QMDS) codes and new quantum error-correcting codes (QECCs) that are better than the best known ones.
The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R=Fp+w1Fp+w2Fp+w22Fp+w1w2Fp+w1w22Fp, where w12−α2=0, w1w2=w2w1, w23−β2w2=0, and α,β∈Fp∖{0}, for a prime p. We begin by introducing a Gray map defined over R, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction.
Abstract In this study, the hull dimensions of ( − 2 v m −1 +1)-constacyclic codes over the ring R = 𝔽 q + v 𝔽 q + · · · + v m −1 𝔽 q with v m = v are determined. In addition, by using a special Gray map, a necessary condition is given under which the generator polynomial of a ( − 2 v m −1 + 1)-constacyclic code over R yields, through the polynomial representation of the Gray map, the generator polynomial of the corresponding cyclic code over F q . Later, an example is presented showing that the parameters of a quantum error-correcting code over F q can be obtained from a ( − 2 v m −1 +1)-constacyclic code over R by means of the X-construction method. Furthermore, the case in which the number of ( − 2 v m −1 + 1)-constacyclic codes of length n over R equals the number of cyclic codes of length 2 n over F q is examined, and an example is provided in which the parameters of the quantum error-correcting codes arising from those cyclic codes—which are Gray images of ( − 2 v m −1 + 1)-constacyclic codes over R —are presented.