The structural properties and construction of quantum codes over using Constacyclic codes over the finite commutative non-chain ring = where are the focus of htis paperand is field having elements with characteristic p where p is an prime such that . A Gray map is defined between and . Decomposing constacyclic codes into cyclic and negacyclic codes over yields the parameters of quantum codes over . Some examples of quantum codes of arbitrary length are also obtained as an application.
The structural properties and construction of quantum codes over using Constacyclic codes over the finite commutative non-chain ring = where are the focus of htis paperand is field having elements with characteristic p where p is an prime such that . A Gray map is defined between and . Decomposing constacyclic codes into cyclic and negacyclic codes over yields the parameters of quantum codes over . Some examples of quantum codes of arbitrary length are also obtained as an application.
This paper is concerned with, structural properties and construction of quantum codes over Z p by using (1+(p−2)ν)-Constacyclic codes over the finite commutative non-chain ring ℜ = Z p +νZ p where ν 2 = ν and Z p is field having p elements with characteristic p where p is prime. A Gray map is defined between ℜ and Z p 2 . The parameters of quantum codes over Z p are obtained by decomposing (1+(p−2)ν)-constacyclic codes into cyclic and negacyclic codes over Z p . As an application, some examples of quantum codes of arbitrary length, are also obtained.
This paper is concerned with, structural properties and construction of quantum codes over Z3 by using constacyclic codes over the finite commutative non-chain ring R = Z3 + νZ3 + ωZ3 + νωZ3 where ν2 = 1, ω2 = ω, νω = νω, and Z3 is field having 3 elements with characteristic 3. A Gray map is defined between R and Z43. The parameters of quantum codes over Z3 are obtained by decomposing constacyclic codes into cyclic and negacyclic codes over Z3. As an application, some examples of quantum codes of arbitrary length, are also obtained.
The structural properties and construction methodologies of quantum codes (QCs) defined over the finite field Z_3, the finite field with three elements, are examined in this work. The primary focus is on utilizing constacyclic codes within a specific algebraic structure the finite commutative non-chain ring R=Z3+uZ3+vZ3+uvZ3, where the indeterminates u and v satisfy the relations u2=1, v2=1, and uv=vu. We provide a set of idempotent generators of the ring and define linear codes and calculated some self-inverse units. The relationship between R and Z34 is established using a gray map. Constacyclic (CC) codes are broken down into cyclic (C) codes and negacyclic (N) codes in order to determine the parameters of QCs over Z_3. Several QCs of arbitrary lengths are developed as an application. In this paper, we calculated those various parameters of QCs which are better than existing parameters of QCs. This paper contains only those units which have self-inverse.
The purpose of this paper is to identify the structural characteristics and construction of quantum codes over Fp using $\alpha_1+\alpha_2 {\mu}+\alpha_3 {\nu}+\alpha_4 {\varpi}+\alpha_5 {\mu}{\nu}+\alpha_6 {\nu}{\varpi}+\alpha_7 {\mu}{\varpi}+\alpha_8 {\mu}{\nu}{\varpi}- $constacyclic codes over ring R=Fp[μ,ν,ϖ]/<μ2−μ,ν2−ν,ϖ2−ϖ,μν−νμ,νϖ−ϖν,μϖ−ϖμ>, here Fp is a finite field with p elements. We define a Gray map from R to Fp8, a distance-preserving map. Breaking down constacyclic codes into cyclic and negacyclic codes results in the creation of quantum codes over the finite field Fp. As an application, some examples are illustrated to obtain the quantum codes of different parameters.
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.
In this paper, we analyze the structural properties and construction technique of quantum codes(QC) over F_l. The focus is on constructing QC using constacyclic codes defined over the finite commutative non-chain ring H_s= F_l+μF_l+νF_l+〖wF〗_l+μνF_l+νwF_l+μwF_l+μν〖wF〗_l, where the indeterminates satisfy μ^2=1, ν^2=ν, w^2=1,μν=νμ,νw=wν,and μw=wμ. A complete set of idempotent generators of the ring H_s is determined, and linear codes are constructed using self-inverse units. A Gray map from H_s to F_l^8is defined to preserve algebraic duality, thereby facilitating the application of the CSS construction method for deriving QC. Constacyclic codes over H_s are decomposed into cyclic and negacyclic codes over F_l. This decomposition enables the explicit computation of the parameters of the resulting QC. Several families of QC of arbitrary length are obtained as illustrative examples. The novelty of this study lies in the exclusive use of self-inverse units in the construction process and in the decomposition behavior of constacyclic codes over H_s for QC construction.