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EPJ Quantum Technology

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

Bo Hyun Kong, Xiying Zheng

Published Feb 6, 2023• 7 citations
Publisher

Abstract

Abstract Let $S_{k}={\mathbb{F}}_{q}[u_{1},u_{2},\ldots ,u_{k}]/\langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 \rangle $ S k = F q [ u 1 , u 2 , … , u k ] / 〈 u i 3 = u i , u i u j = u j u i = 0 〉 , where $1\leq i,j\leq k$ 1 ≤ i , j ≤ k , $q=p^{m}$ q = p m , p is an odd prime. First, we define two new Gray maps $\phi _{k}$ ϕ k and $\varphi _{k}$ φ k , and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over $\mathbb{F}_{q}$ F q by using CSS construction, and compare the constructed codes better than the existing literature.

DOI
10.1140/epjqt/s40507-023-00160-7

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Abstract Let Sk=Fq[u1,u2,…,uk]/⟨ui3=ui,uiuj=ujui=0⟩S_{k}={\mathbb{F}}_{q}[u_{1},u_{2},\ldots ,u_{k}]/\langle u^{3}_{i}=u_{i},u_{i}u_{j}=u_{j}u_{i}=0 \rangle Sk​=Fq​[u1​,u2​,…,uk​]/⟨ui3​=ui​,ui​uj​=uj​ui​=0⟩ S k = F q [ u 1 , u 2 , … , u k ] / 〈 u i 3 = u i , u i u j = u j u i = 0 〉 , where 1≤i,j≤k1\leq i,j\leq k1≤i,j≤k 1 ≤ i , j ≤ k , q=pmq=p^{m}q=pm q = p m , p is an odd prime. First, we define two new Gray maps ϕk\phi _{k}ϕk​ ϕ k and φk\varphi _{k}φk​ φ k , and study their Gray images. Further, we determine the structure of constacyclic codes and their dual codes, and give a necessary and sufficient conditions of constacyclic codes to contain their duals. Finally, we obtain some new quantum codes over Fq\mathbb{F}_{q}Fq​ F q by using CSS construction, and compare the constructed codes better than the existing literature.
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In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm\mathbb{F}_{p^m}Fpm​.
This paper primarily investigates the structural properties of constacyclic codes over the ring , defined as where i, j = 1, 2, 3, ..., k, and k, m are positive integers. Here, denotes a finite field of order pm with characteristic p, an odd prime. Furthermore, we determine the necessary and sufficient conditions for the duals of constacyclic codes to exist. These findings make it easier to create new Quantum Error-Correcting (QEC) codes throughout the ring (i.e., when k = 1), as well as optimal linear codes that make use of the Gray images of constacyclic codes. Additionally, Table 4 presents several Linear Complementary Dual (LCD) codes obtained using the Gray map.