Finite Non-chain Ring Research Articles

Some Results for Double Cyclic Codes over Fq+vFq+v2Fq.

Let Fq be a finite field with an odd characteristic. In this paper, we present a new result about double cyclic codes over a finite non-chain ring. Specifically, we study the double cyclic code over Fq+vFq+v2Fq with v3=v, which is isomorphic to Fq×Fq×Fq. This study mainly involves generator polynomials and generator matrices. The generating polynomial of the dual code is also obtained. We show the relationship between the generating polynomials of the double cyclic codes and those of their dual codes. Finally, as an application of these results, we construct some optimal codes over F3.

New quantum codes from constacyclic codes over a general non-chain ring

Let [Formula: see text] be a prime power and let [Formula: see text] be a finite non-chain ring, where [Formula: see text], are polynomials, not all linear, which split into distinct linear factors over [Formula: see text]. We characterize constacyclic codes over the ring [Formula: see text] and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials [Formula: see text], [Formula: see text], is irrelevant while constructing quantum codes from constacyclic codes over [Formula: see text], it depends only on their degrees.

New MDS EAQECCs derived from constacyclic codes over [formula omitted

Entanglement-assisted quantum error-correcting codes (EAQECCs) can improve information rate by using pre-shared entanglement between the sender and the receiver. In this paper, we construct three new infinite families of maximum distance separable (MDS) EAQECCs with less pre-shard maximally entangled states based on the Gray images of constacyclic codes over the finite non-chain ring Fq2+vFq2 with v2=v. The parameters of these MDS EAQECCs are not covered by the codes available in the previous literatures. Furthermore, our results generalize and improve the work of Lu et al. (2018) [16], (2020) [19].

Quantum and LCD codes from skew constacyclic codes over a finite non-chain ring

Quantum and LCD codes from skew constacyclic codes over a finite non-chain ring

Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring

Construction of LCD and new quantum codes from cyclic codes over a finite non-chain ring

Quantum codes from Hermitian dual-containing constacyclic codes over $${\mathbb {F}}_{q^{2}}+{v}{\mathbb {F}}_{q^{2}}$$

Let $${\mathbb {R}}$$ be the finite non-chain ring $${\mathbb {F}}_{{ q}^{2}}+{v}{\mathbb {F}}_{{ q}^{2}}$$ , where $${v}^{2}={v}$$ and q is an odd prime power. In this paper, we study quantum codes over $${\mathbb {F}}_{{ q}}$$ from constacyclic codes over $${\mathbb {R}}$$ . We define a class of Gray maps, which preserves the Hermitian dual-containing property of linear codes from $${\mathbb {R}}$$ to $${\mathbb {F}}_{{ q}^{2}}$$ . We study $${\alpha }(1-2v)$$ -constacyclic codes over $${\mathbb {R}}$$ , and show that the images of $$\alpha (1-2v)$$ -constacyclic codes over $${\mathbb {R}}$$ under the special Gray map are $$\alpha ^{2}$$ -constacyclic codes over $${\mathbb {F}}_{{ q}^{2}}$$ . Some new non-binary quantum codes are obtained via the Gray map and the Hermitian construction from Hermitian dual-containing $$\alpha (1-2v)$$ -constacyclic codes over $${\mathbb {R}}$$ .

(σ, δ)-Skew quasi-cyclic codes over the ring $\mathbb {Z}_{4}+u\mathbb {Z}_{4}$

Let $R=\mathbb {Z}_{4}+u\mathbb {Z}_{4}$ be a finite non-chain ring, where u2 = 1. In this paper, we consider (σ, δ)-skew quasi-cyclic codes over the ring R, where σ is an automorphism of R and δ is an inner σ-derivation of R. We determine the structure of 1-generator (σ, δ)-skew quasi-cyclic codes over R and give a sufficient condition for 1-generator (σ, δ)-skew quasi-cyclic codes over R to be free. We also determine a distance bound for free 1-generator (σ, δ)-skew quasi-cyclic codes. Moreover, using the residue codes of these codes over R we obtain some good $\mathbb {Z}_{4}$ -linear codes. Finally, we give the characterization of Euclidean dual codes of (σ, δ)-skew quasi-cyclic codes.

Bounds on Covering Radius of Some Codes Over F₂ + uF₂ + vF₂ + uvF₂

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R=\mathbb {F}_{2}+u\mathbb {F}_{2}+v\mathbb {F}_{2}+uv\mathbb {F}_{2}$ </tex-math></inline-formula> be a finite non-chain ring, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u^{2}=u$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v^{2}=v$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$uv=vu$ </tex-math></inline-formula> . We give the lower and upper bounds on the covering radius of different types of repetition codes for Chinese Euclidean distance over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> . Furthermore, we determine the upper bound on the covering radius of block repetition codes, simplex codes of types <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> , MacDonald codes of types <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> for Chinese Euclidean distance over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> .

Construction of new quantum codes derived from constacyclic codes over $${\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+\cdots +u^{r-1}{\mathbb {F}}_{q^2}$$

Let $$R={\mathbb {F}}_{q^2}+u{\mathbb {F}}_{q^2}+\cdots +u^{r-1}{\mathbb {F}}_{q^2}$$ be a finite non-chain ring, where q is a prime power, $$u^{r}=1$$ and $$r|(q+1)$$. In this paper, we study u-constacyclic codes over the ring R. Using the matrix of Fourier transform, a Gray map from R to $${\mathbb {F}}_{q^2}^{r}$$ is given. Under the special Gray map, we show that the image of Gray map of u-constacyclic codes over R are cyclic codes over $${\mathbb {F}}_{q^2}$$, and some new quantum codes are obtained via the Gray map and Hermitian construction from Hermitian dual-containing u-constacyclic codes.

Polyadic constacyclic codes over a non-chain ring $$\mathbb {F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle $$

Let f(u) and g(v) be two polynomials, not both linear, which split into distinct linear factors over $$\mathbb {F}_{q}$$. Let $$\mathcal {R}=\mathbb {F}_{q}[u,v]/ \langle f(u),g(v),uv-vu\rangle $$ be a finite commutative non-chain ring. In this paper, we study polyadic $$\lambda $$-constacyclic codes of Type I and Type II over $$\mathcal {R}$$ for $$\lambda \in \mathbb {F}_q^*$$. The Gray images of polyadic negacyclic codes and their extensions lead to construction of self-dual, isodual, self-orthogonal and complementary dual(LCD) codes over $$\mathbb {F}_q$$. We also study $$\alpha $$-constacyclic codes over $$\mathcal {R}$$ for any unit $$\alpha $$ in $$\mathcal {R}$$.

MacDonald codes over the ring $${\mathbb {F}}_{p}+v{\mathbb {F}}_{p}+v^2{\mathbb {F}}_{p}$$

In this paper, we consider MacDonald codes over the finite non-chain ring $${\mathbb {F}}_p+v{\mathbb {F}}_p+v^2{\mathbb {F}}_p$$ and their applications in constructing secret sharing schemes and association schemes, where p is an odd prime and $$v^3=v$$ . We give some structural properties of MacDonald codes first. Then, we study the weight enumerators of torsion codes of these MacDonald codes. As some applications, constructing secret sharing schemes and association schemes is also investigated.

Two-Weight Linear Codes and Their Applications in Secret Sharing

Linear codes with few weighs have many applications in secret sharing. Determining the access structure of the secret sharing scheme based on a linear code is a very difficult problem. We provides a method to construct a class of two-weight torsion codes over finite non-chain ring. We determine the minimal codewords of these torsion codes over the finite non-chain ring. Based on the two-weight codes, we find the access structures of secret sharing schemes.

A note on skew constacyclic codes over 𝔽q + u𝔽q + v𝔽q

In this paper, the skew constacyclic codes over finite non-chain ring [Formula: see text], where [Formula: see text], [Formula: see text] is an odd prime and [Formula: see text], are studied. We show that the Gray image of a skew [Formula: see text]-constacyclic code of length [Formula: see text] over [Formula: see text] is a skew quasi-twisted code of length [Formula: see text] over [Formula: see text] of index 3. Further, the structural properties of skew constacyclic codes over [Formula: see text] are obtained by the decomposition method.

Duadic negacyclic codes over a finite non-chain ring

Let [Formula: see text] be a polynomial of degree [Formula: see text] which splits into distinct linear factors over a finite field [Formula: see text]. Let [Formula: see text] be a finite non-chain ring. In an earlier paper, we studied duadic and triadic codes over [Formula: see text] and their Gray images. Here, we study duadic negacyclic codes of Type I and Type II over the ring [Formula: see text], their extensions and their Gray images. As a consequence some self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over [Formula: see text] are constructed. Some examples are also given to illustrate this.

Quantum codes from (1 − 2v)-constacyclic codes over the ring 𝔽q + u𝔽q + v𝔽q + uv𝔽q

In this paper, structural properties of [Formula: see text]-constacyclic codes over the finite non-chain ring [Formula: see text] are studied, where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a power of some odd prime. As an application, some better quantum codes, compared with previous work, are obtained.

Constacyclic codes over the ring $${\mathbb {F}}_q+v{\mathbb {F}}_q+v^{2}{\mathbb {F}}_q$$ F q + v F q + v 2 F q and their applications of constructing new non-binary quantum codes

Let $$R={\mathbb {F}}_q+v{\mathbb {F}}_q+v^{2}{\mathbb {F}}_q$$ be a finite non-chain ring, where q is an odd prime power and $$v^3=v$$ . In this paper, we propose two methods of constructing quantum codes from $$(\alpha +\beta v+\gamma v^{2})$$ -constacyclic codes over R. The first one is obtained via the Gray map and the Calderbank–Shor–Steane construction from Euclidean dual-containing $$(\alpha +\beta v+\gamma v^{2})$$ -constacyclic codes over R. The second one is obtained via the Gray map and the Hermitian construction from Hermitian dual-containing $$(\alpha +\beta v+\gamma v^{2})$$ -constacyclic codes over R. As an application, some new non-binary quantum codes are obtained.

Duadic and triadic codes over a finite non-chain ring and their Gray images

Let f(u) be a polynomial of degree m, m ≥ 2, which splits into distinct linear factors over a finite field &#x1d53d;q. Let &#x1d4e1; = &#x1d53d;q[u] / ⟨ f(u) ⟩ be a finite non-chain ring. In this paper, we study duadic codes, their extensions and triadic codes over the ring &#x1d4e1;. A Gray map from &#x1d4e1;n to (&#x1d53d;q m)n is defined which preserves self-duality of linear codes. As a consequence, self-dual, isodual, self-orthogonal and complementary dual(LCD) codes over &#x1d53d; q are constructed. Some examples are also given to illustrate this.

Duadic and triadic codes over a finite non-chain ring and their Gray images

Duadic and triadic codes over a finite non-chain ring and their Gray images

On quantum codes obtained from cyclic codes over 𝔽2 + u𝔽2 + u2𝔽2

In the paper, we consider a finite non-chain commutative ring [Formula: see text], where [Formula: see text]. We mainly study the construction of quantum codes from cyclic codes over [Formula: see text]. For this, we obtained self-orthogonal codes over [Formula: see text] as gray images of linear and cyclic codes over [Formula: see text], then these codes over [Formula: see text] correspond to a cyclic code over [Formula: see text] of odd length [Formula: see text] used to determine the parameters of the quantum codes.

Good Linear Codes from Quadratic Residue Codes over a Finite Non‐Chain Ring

This paper is devoted to the study of quadratics residue codes and their extended codes over a finite non-chain ring. A class of Gray maps preserving the self-duality from the ring to the finite field are introduced. Some structural properties of quadratic residue codes and their extended codes are given. As an interesting application of these families of codes, some good linear codes are obtained by a special Gray map.