Non-chain Ring Research Articles

Skew Constacyclic Codes over a Non-Chain Ring.

In this paper, we investigate the algebraic structure of the non-local ring Rq=Fq[v]/⟨v2+1⟩ and identify the automorphisms of this ring to study the algebraic structure of the skew constacyclic codes and their duals over this ring. Furthermore, we give a necessary and sufficient condition for the skew constacyclic codes over Rq to be linear complementary dual (LCD). We present some examples of Euclidean LCD codes over Rq and tabulate the parameters of Euclidean LCD codes over finite fields as the Φ-images of these codes over Rq, which are almost maximum distance separable (MDS) and near MDS. Eventually, by making use of Hermitian linear complementary duals of skew constacyclic codes over Rq and the map Φ, we give a class of entanglement-assisted quantum error correcting codes (EAQECCs) with maximal entanglement and tabulate parameters of some EAQECCs with maximal entanglement over finite fields.

Linear Codes and Linear Complementary Pairs of Codes Over a Non-Chain Ring

Let [Formula: see text] be an odd prime number, [Formula: see text] for a positive integer [Formula: see text], let [Formula: see text] be the finite field with [Formula: see text] elements and [Formula: see text] be a primitive element of [Formula: see text]. We first give an orthogonal decomposition of the ring [Formula: see text], where [Formula: see text] and [Formula: see text] for a fixed integer [Formula: see text]. In addition, Galois dual of a linear code over [Formula: see text] is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring [Formula: see text] are investigated as well. Remarkably, we obtain that if linear codes [Formula: see text] and [Formula: see text] are a complementary pair, then the code [Formula: see text] and the dual code [Formula: see text] of [Formula: see text] are equivalent to each other.

QUANTUM CODES OBTAINED THROUGH -CONSTACYCLIC CODES OVER

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.

QEC and EAQEC codes from cyclic codes over non-chain rings

QEC and EAQEC codes from cyclic codes over non-chain rings

A study of quantum codes obtained from cyclic codes over a non-chain ring

A study of quantum codes obtained from cyclic codes over a non-chain ring

Few-weight codes over a non-chain ring associated with simplicial complexes and their distance optimal Gray image

In this paper, we construct new infinite families of codes over the ring R=F2+uF2+vF2+uvF2,u2=v2=0,uv=vu by employing simplicial complexes. When the simplicial complexes are all generated by a single maximal element, we compute the Lee weight distributions of these classes of codes. By the Gray map, we show that some classes of codes are minimal and some classes of these codes are distance optimal.

DNA codes over finite local Frobenius non-chain rings of length 5 and nilpotency index 4

Abstract A one to one correspondence between the elements of a finite local Frobenius non-chain ring of length 5 and nilpotency index 4, and k-tuples of DNA codewords is established. Using this map the structure of DNA codes over these rings is determined, the length of the code is relatively prime to the characteristic of the residue field of the ring.

An infinite family of Griesmer quasi-cyclic self-orthogonal codes

Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field Fpm. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring Rp,m=Fpm[u,v]/〈u2=v2=0,uv=vu〉, where m is a positive integer, α=a+ub+vc+uvd is a unit of Rp,m, a,b,c,d∈Fpm, and a is nonzero. We then find a Gray map from Rp,m[x]/〈xn−α〉 (with respect to homogeneous weights) to Fpm[x]/〈xp3m+1n−a〉 (with respect to Hamming weights), which is linear and preserves minimum weights. We present an efficient algorithm for finding the Gray images of α-constacyclic codes over Rp,m of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over Fpm of length p3m+1 and index p3m. In particular, some family turns out to be “Griesmer” codes; these Griesmer quasi-cyclic self-orthogonal codes are “new” codes compared with previously known Griesmer codes of dimension 4.

A family of constacyclic codes over a class of non-chain rings $${\mathcal {A}}_{q,r}$$ and new quantum codes

A family of constacyclic codes over a class of non-chain rings $${\mathcal {A}}_{q,r}$$ and new quantum codes

New Quantum and LCD Codes over Finite Fields of Even Characteristic

For an integer m ≥ 1, we study cyclic codes of length l over a commutative non-chain ring F2m + uF2m , where u2 = u . With a new Gray map and Euclidean dual-containing cyclic codes, we provide many new and superior codes to the best-known quantum error-correcting codes. Also, we characterise LCD codes of length l with respect to their generator polynomials and prove that F2m − image of an LCD code of length l is an LCD code of length 2l . Finally, we provide several optimal LCD codes from the Gray images of LCD codes over F2m + uF2m .

Quantum Codes Obtained through Constacyclic Codes over Z3 + νZ3 + ωZ3 + νωZ3

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.

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

New Quantum Codes from Skew Constacyclic Codes Over a Class Of Non-Chain Rings Re, q

New Quantum Codes from Skew Constacyclic Codes Over a Class Of Non-Chain Rings Re, q

Cyclic codes over a non-chain ring Re,q and their application to LCD codes

Let Fq be a finite field of order q, a prime power integer, such that q=et+1 where t≥1,e≥2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring Re,q=Fq[u]/〈ue−1〉. We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal Fq-linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when gcd⁡(n,q)≠1 and gcd⁡(n,q)=1, respectively. It is proved that a cyclic code C of length n is an LCD code if and only if its Gray image φ(C) is an LCD code of length en over Fq. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over Re,q.

Self dual, reversible and complementary duals constacyclic codes over finite local Frobenius non-chain rings of length 5 and nilpotency index 4.

Abstract Over finite local Frobenius non-chain rings of length 5 and nilpotency index 4 and when the length of the code is relatively prime to the characteristic of the residue field of the ring, the structure of the dual of γ-constacyclic codes is established and the algebraic characterization of self-dual, reversible γ-constacyclic codes and γ-constacyclic codes with complementary dual are given.

DNA codes over finite local Frobenius non-chain rings of length 4

A one to one correspondence between the elements of a finite local Frobenius non-chain ring of length 4 and k-tuples of DNA codewords is established. Using this map the structure of DNA codes over finite local Frobenius non-chain rings of length 4 is determined, the length of the code is relatively prime to the characteristic of the residue field of the ring.

Character sums over a non-chain ring and their applications

Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain 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 $${\mathbb {R}}$$ .

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

Let p be a prime of the form $$p=mt+1$$ , where integers $$t\ge 1, m\ge 2$$ and $$R_m=\mathbb {F}_p[u]/\langle u^m-1\rangle .$$ Thus, $$R_m$$ is a finite commutative non-chain ring. For a given unit $$\lambda \in R_m$$ , we study $$\lambda $$ -constacyclic codes of length n over $$R_m$$ . The necessary and sufficient conditions for these codes to contain their Euclidean duals are determined. As an application from dual-containing $$\lambda $$ -constacyclic codes over $$R_m$$ , for $$m=2,3,4$$ , we obtain many new quantum codes that improve on the known existing quantum codes.

(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.