Non-chain Ring Research Articles

Polyadic Cyclic Codes over a Non-Chain Ring

Let f(u) and g(v) be two polynomials of degree k and l respectively, not both linear which split into distinct linear factors over Fq. Let be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring R. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from which preserves duality. The Gray images of polyadic codes and their extensions over the ring R lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over Fq. Some examples are also given to illustrate this.

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> .

Quantum codes obtained through (1+(p−2)ν)-constacyclic codes over Zp+νZp

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.

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

Abstract The family of finite local Frobenius non-chain rings of length 5 and nilpotency index 4 is determined, as a by-product all finite local Frobenius non-chain rings with p 5 elements (p a prime) and nilpotency index 4 are given. And the number and structure of γ-constacyclic codes over those rings, of length relatively prime to the characteristic of the residue field of the ring, are determined.

Skew 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 general skew cyclic codes and $$\theta _t$$ -skew constacyclic codes over the ring $${\mathcal {R}}$$ where $$\theta _t$$ is an automorphism of $${\mathcal {R}}$$ .

Cyclic codes over $${\mathbb {F}}_2 +u{\mathbb {F}}_2+v{\mathbb {F}}_2 +v^2 {\mathbb {F}}_2 $$ with respect to the homogeneous weight and their applications to DNA codes

In this paper, we study cyclic codes and their duals over the local Frobenius non-chain ring $$R={\mathbb {F}}_2[u,v] / \langle u^2=v^2,uv \rangle $$, and we obtain optimal binary linear codes with respect to the homogeneous weight over R via a Gray map. Moreover, we characterize DNA codes as images of cyclic codes over R.

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.

Linear Codes Over a Non-Chain Ring and the MacWilliams Identities

This paper is concerned with the linear codes over the non-chain ring $R=\mathbb {F}_{2}[v]/\langle v^{4}-v\rangle $ . First, several weight enumerators over $R$ are defined. Then the MacWilliams identity is obtained, which can establish an important relation respect to the complete weight enumerators. Meanwhile, the symmetric weight enumerators between linear code and its dual over $R$ are established by the Gray map from $R^{n}$ to $\mathbb {F}_{2}^{4n}$ . Finally, several examples are given to illustrate our main results and some open problems are also proposed.

Linear Codes over the Finite Ring &amp;lt;i&amp;gt;Z&amp;lt;/i&amp;gt;&amp;lt;sub&amp;gt;15&amp;lt;/sub&amp;gt;

In this paper, the structure of the non-chain ring Z15 is studied. The ideals of the ring Z15 are obtained through its non-units and the Lee weights of elements in Z15 are presented. On this basis, by the Chinese Remainder Theorem, we construct a unique expression of an element in Z15. Further, the Gray mapping from Zn15 to Z2n15 is defined and it’s shown to be distance preserved. The relationship between the minimum Lee weight and the minimum Hamming weight of the linear code over the ring Z15 is also obtained and we prove that the Gray map of the linear code over the ring Z15 is also linear.

Infinite families of MDR cyclic codes over [formula omitted] via constacyclic codes over [formula omitted

We study α-constacyclic codes over the Frobenius non-chain ring R≔Z4[u]∕〈u2−1〉 for any unit α of R. We obtain new MDR cyclic codes over Z4 using a close connection between α-constacyclic codes over R and cyclic codes over Z4. We first explicitly determine generators of all α-constacyclic codes over R of odd length n for any unit α of R. We then explicitly obtain generators of cyclic codes over Z4 of length 2n by using a Gray map associated with the unit α. This leads to a construction of infinite families of MDR cyclic codes over Z4, where a MDR code means a maximum distance with respect to rank code in terms of the Hamming weight or the Lee weight. We obtain 202 new cyclic codes over Z4 of lengths 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50 and 54 by implementing our results in Magma software; some of them are also MDR codes with respect to the Hamming weight or the Lee weight.

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.

On Self‐dual and LCD Double Circulant Codes over a Non‐chain Ring*

In this paper, we study double circulant codes of length 2n over the non-chain ring R = 𝔽 q + v𝔽 q + v 2𝔽 q ; where q is an odd prime power and v 3 = v. Exact enumerations of self-dual and LCD double circulant codes of length 2n over R are derived. When n is an odd prime, using random coding, we obtain families of asymptotically good self-dual and LCD codes of length 6n over 𝔽 q .

Skew constacyclic codes over the local Frobenius non-chain rings of order 16

We introduce skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over some of these rings.

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.

The dual of a constacyclic code, self dual, reversible constacyclic codes and constacyclic codes with complementary duals over finite local Frobenius non-chain rings with nilpotency index 3

Over finite local Frobenius non-chain rings with nilpotency index 3 and when the length of the codes 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 γ-constacyclic codes, reversible γ-constacyclic codes and γ-constacyclic codes with complementary dual are given. Generators for the dual code are obtained from those of the original constacyclic code.

On codes over Frobenius rings: generating characters, MacWilliams identities and generator matrices

Codes over commutative Frobenius rings are studied with a focus on local Frobenius rings of order 16 for illustration. The main purpose of this work is to present a method for constructing a generating character for any commutative Frobenius ring. Given such a character, the MacWilliams identities for the complete and symmetrized weight enumerators can be easily found. As examples, generating characters for all commutative local Frobenius rings of order 16 are given. In addition, a canonical generator matrix for codes over local non-chain rings is discussed. The purpose is to show that when working over local non-chain rings, a canonical generator matrix exists but is less than useful which emphases the difficulties in working over such rings.

On a class of constacyclic codes of length 4ps over 𝔽pm + u𝔽pm

Let [Formula: see text] be a prime such that [Formula: see text]. For any unit [Formula: see text] of [Formula: see text], we determine the algebraic structures of [Formula: see text]-constacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text], [Formula: see text]. If the unit [Formula: see text] is a square, each [Formula: see text]-constacyclic code of length [Formula: see text] is expressed as a direct sum of an -[Formula: see text]-constacyclic code and an [Formula: see text]-constacyclic code of length [Formula: see text] If the unit [Formula: see text] is not a square, then [Formula: see text] can be decomposed into a product of two irreducible coprime quadratic polynomials which are [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. By showing that the quotient rings [Formula: see text] and [Formula: see text] are local, non-chain rings, we can compute the number of codewords in each of [Formula: see text]-constacyclic codes. Moreover, the duals of such codes are also given.

On the Structure and Distances of Repeated-Root Constacyclic Codes of Prime Power Lengths Over Finite Commutative Chain Rings

Let p be a prime, s be a positive integer, and R be a finite commutative chain ring with the characteristic as a power of p. For a unit λ ε R, λ-constacyclic codes of length ps over R are ideals of the quotient ring R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> -λ). In this paper, we derive necessary and sufficient conditions under which the quotient ring R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a chain ring. When R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a chain ring, all λ-constacyclic codes of length ps over R are known. In this paper, we establish the algebraic structures of all λ-constacyclic codes of length ps over R when R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a non-chain ring. We also determine the number of codewords in each of these codes. Using their algebraic structures, we obtain symbol-pair distances, Rosenbloom-Tsfasman (RT) distances, and RT weight distributions of all constacyclic codes of length ps over R. Apart from this, we derive necessary and sufficient conditions under which a constacyclic code of length ps over R is maximumdistance separable with respect to the: 1) Hamming metric; 2) symbol-pair metric; and 3) RT metric. We also provide an algorithm to decode the constacyclic codes of length ps over R using the known decoding algorithms of linear codes over finite fields with respect to the Hamming, symbol-pair, and RT metrics.