Finite Commutative Chain Ring Research Articles

Determinants of Arrowhead Matrices over Finite Commutative Chain Rings

Arrowhead matrices have attracted attention due to their rich algebraic structures and numerous applications. In this paper, we focus on the enumeration of n × n arrowhead matrices with prescribed determinant over a finite field Fq and over a finite commutative chain ring R. The number of n × n arrowhead matrices over Fq of a fixed determinant a is determined for all positiveintegers n and for all elements a ∈ Fq. As applications, this result is used in the enumeration of n × n non-singular arrowhead matrices with prescribed determinant over R. Subsequently, some bounds on the number of n × n singular arrowhead matrices over R of a fixed determinant are given. Finally, some open problems are presented.

On Hamming distance distributions of repeated-root constacyclic codes of length 3ps over [formula omitted

Let p≠3 be any prime. The algebraic structures of all λ-constacyclic codes of length 3ps over the finite commutative chain ring R=Fpm+uFpm are provided by classifying all ideals of the local ring R[x]〈x3ps−λ〉. Using these structures, in this paper, the exact values of Hamming distances of all such λ-constacyclic codes of length 3ps are established. As an application, we identify all maximal distance separable λ-constacyclic codes of length 3ps.

Submodule codes as spherical codes in buildings

We give a generalization of subspace codes by means of codes of modules over finite commutative chain rings. We define a new class of Sperner codes and use results from extremal combinatorics to prove the optimality of such codes in different cases. Moreover, we explain the connection with Bruhat–Tits buildings and show how our codes are the buildings’ analogue of spherical codes in the Euclidean sense.

On J-Diagrams for the One Groups of Finite Chain Rings

Let R be a finite commutative chain ring with invariants p,n,r,k,m. The purpose of this article is to study j-diagrams for the one group H=1+J(R) of R, where J(R)=(π) is Jacobson radical of R. In particular, we prove the existence and uniqueness of j-diagrams for such one group. These j-diagrams help us to solve several problems related to chain rings such as the structure of their unit groups and a group of all symmetries of {πk′}, where k′∣k. The invariants p,n,r,k,m and the Eisenstein polynomial by which R is constructed over its Galois subring determine fully the j-diagram for H.

Double Constacyclic Codes Over Two Finite Commutative Chain Rings

Many kinds of codes which possess two cycle structures over two special finite commutative chain rings, such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ </tex-math></inline-formula> -additive cyclic codes and quasi-cyclic codes of fractional index etc., were proved asymptotically good. In this paper we extend the study in two directions: we consider any two finite commutative chain rings with a surjective homomorphism from one to the other, and consider double constacyclic structures. We construct an extensive kind of double constacyclic codes over two finite commutative chain rings. And, developing a probabilistic method suitable for quasi-cyclic codes over fields, we prove that the double constacyclic codes over two finite commutative chain rings are asymptotically good.

On Symbol-Pair Distance of a Class of Constacyclic Codes of Length 3ps over Fpm+uFpm

Let p≠3 be any prime. In this paper, we compute symbol-pair distance of all γ-constacyclic codes of length 3ps over the finite commutative chain ring R=Fpm+uFpm, where γ is a unit of R which is not a cube in Fpm. We give the necessary and sufficient condition for a symbol-pair γ-constacyclic code to be an MDS symbol-pair code. Using that, we provide all MDS symbol-pair γ-constacyclic codes of length 3ps over R. Some examples of the symbol-pair distance of γ-constacyclic codes of length 3ps over R are provided.

Convolutional codes over finite chain rings, MDP codes and their characterization

&lt;p style='text-indent:20px;'&gt;In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate (reverse) MDP convolutional codes over a finite chain ring with (reverse) MDP convolutional codes over its residue field. Finally, we provide a construction of (reverse) MDP convolutional codes over finite chain rings generalizing the notion of (reverse) superregular matrices.&lt;/p&gt;

Asymptotically good $ \mathbb{Z}_{p}\mathbb{Z}_{p}[u]/\langle u^{t}\rangle $-additive cyclic codes

$ \mathbb{Z}_{2}(\mathbb{Z}_{2}+u\mathbb{Z}_{2}) $-additive cyclic codes were proved to be asymptotically good in 2020 by Yao et al., where $ u^{2} = 0 $. We extend the study to the double cyclic codes over two finite commutative chain rings. Let $ R_{t} = \mathbb{Z}_{p}[u]/\langle u^{t}\rangle = \mathbb{Z}_{p}+u\mathbb{Z}_{p}+u^{2}\mathbb{Z}_{p}+\ldots+u^{t-1}\mathbb{Z}_{p} $ be a chain ring, where $ u^{t} = 0 $. We construct a class of $ \mathbb{Z}_{p}R_{t} $-additive cyclic codes generated by pairs of polynomials, where $ p $ is a prime number. By using probabilistic methods, we study the asymptotic behaviour of the rates and relative minimum distances of a certain class of the codes. We show that there exists an asymptotically good infinite sequence of $ \mathbb{Z}_{p}R_{t} $-additive cyclic codes with the relative minimum distance of the code is convergent to $ \delta $, and the rate is convergent to $ \frac{1}{1+p^{t-1}} $ for $ 0&lt; \delta&lt; \frac{1}{1+p^{t-1}} $, and $ t\geq1 $.

On codes over product of finite chain rings

In this paper, codes over the direct product of two finite commutative chain rings are studied. The standard form of the parity-check matrix is determined. The structure of self-dual codes is described. A distance preserving Gray map from the direct product of chain rings to a finite field is defined. Two upper bounds on minimum distance are obtained.

Factoring the Dedekind-Frobenius determinant of a semigroup

The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutative chain ring. The former was motivated by combinatorics and the latter by coding theory over finite rings. We prove that the algebra of the multiplicative semigroup of a finite Frobenius ring is Frobenius over any field whose characteristic doesn't divide that of the ring. As a consequence we obtain an easier proof of Kovács's theorem that the algebra of the monoid of matrices over a finite field is a direct product of matrix algebras over group algebras of general linear groups (outside of the characteristic of the finite field).

On the enumeration and classification of σ-LCD codes over finite commutative chain rings

Let Re denote a finite commutative chain ring with the maximal ideal 〈u〉 and the nilpotency index e, and let R‾e=Re/〈u〉 be the residue field of Re. Let σ0 be an automorphism of Re, and let σ‾0 be the corresponding automorphism of the residue field R‾e of Re, defined as σ‾0(a+〈u〉)=σ0(a)+〈u〉 for all a+〈u〉∈R‾e. Let σ be an automorphism of Ren corresponding to the automorphism σ0 of Re, defined as σ(v1,v2,…,vn)=(σ0(v1),σ0(v2),…,σ0(vn)) for all (v1,v2,…,vn)∈Ren. In this paper, we obtain explicit enumeration formulae for all σ-LCD codes of an arbitrary length over the chain ring Re when σ‾02 is the identity automorphism of R‾e. With the help of these enumeration formulae and by applying the classification algorithm, we classify all Euclidean LCD codes of lengths 2, 3, 4 and 5 over the chain ring F2[u]/〈u2〉 and of lengths 2, 3 and 4 over the chain ring F3[u]/〈u2〉, and all σ-LCD codes of lengths 2, 3 and 4 over the chain ring F4[u]/〈u2〉, where σ0 is an automorphism of F4[u]/〈u2〉 such that the corresponding automorphism σ‾0 of the residue field F4 has order 2. Besides this, we show that the class of σ-LCD codes over Re is asymptotically good, and that every free linear [n,k,d]-code over Re is equivalent to a σ-LCD [n,k,d]-code over Re when |R‾e|>4. We also explicitly determine all inequivalent σ-LCD [n,1,d]-codes and [n,n−1,d]-codes over Re for 1≤d≤n.

Classification of chain rings

&lt;abstract&gt;&lt;p&gt;An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $ p, n, r, k, k', m $ up to isomorphism by finite commutative chain rings ($ k' = 1 $). Thus the classification of chain rings is reduced to that of finite commutative chain rings.&lt;/p&gt;&lt;/abstract&gt;

On classification of finite commutative chain rings

&lt;abstract&gt;&lt;p&gt;Let $ R $ be a finite commutative chain ring with invariants $ p, n, r, k, m. $ It is known that $ R $ is an extension over a Galois ring $ GR(p^n, r) $ by an Eisenstein polynomial of some degree $ k $. If $ p\nmid k, $ the enumeration of such rings is known. However, when $ p\mid k $, relatively little is known about the classification of these rings. The main purpose of this article is to investigate the classification of all finite commutative chain rings with given invariants $ p, n, r, k, m $ up to isomorphism when $ p\mid k. $ Based on the notion of j-diagram initiated by Ayoub, the number of isomorphism classes of finite (complete) chain rings with $ (p-1)\nmid k $ is determined. In addition, we study the case $ (p-1)\mid k, $ and show that the classification is strongly dependent on Eisenstein polynomials not only on $ p, n, r, k, m. $ In this case, we classify finite (incomplete) chain rings under some conditions concerning the Eisenstein polynomials. These results yield immediate corollaries for p-adic fields, coding theory and geometry.&lt;/p&gt;&lt;/abstract&gt;

Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, D2n be the dihedral group of size 2n and R[D2n] be the dihedral group ring. In this paper, we completely characterize left ideals of R[D2n] (called left D2n-codes) when gcd(char(R),n)=1. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over R×S, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where R=Fpm is a Galois field, we give a classification for left D2N-codes over Fpm, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.

Constacyclic Codes over Finite Chain Rings of Characteristic p

Let R be a finite commutative chain ring of characteristic p with invariants p,r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/&lt;xN−λ&gt; and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k=2 the exact number of such codes is provided.

Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over \({{\mathbb{F}_{p^m}[u]}/{\langle u^3\rangle}}\)

Let p be a prime, s, m be positive integers, γ be a nonzero element of the finite field Fpm, and let R=Fpm[u]/⟨u3⟩ be the finite commutative chain ring. In this paper, the symbol-pair distances of all γ-constacyclic codes of length ps over R are completely determined.

Complete classification of repeated-root [formula omitted]-constacyclic codes of prime power length over [formula omitted

Let R=Fpm[u]∕〈u3〉 be the finite commutative chain ring with unity, where p is a prime, m is a positive integer and Fpm is a finite field with pm elements. In this study, we investigate σ-constacyclic codes of length ps over R, that is, ideals of the ring R[x]∕〈xps−σ〉, where σ is a nonzero element of the field Fpm. First, we classify all cyclic codes of length ps over R and obtain the number of codewords in each type of those cyclic codes. Furthermore, by using the ring isomorphism we completely determine the structure of σ-constacyclic codes of length ps over R.

Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring

A finite ring with an identity whose lattice of ideals forms a unique chain is called a finite chain ring. Let R be a commutative chain ring with invariants p,n,r,k,m. It is known that R is an Eisenstein extension of degree k of a Galois ring S=GR(pn,r). If p−1 does not divide k, the structure of the unit group U(R) is known. The case (p−1)∣k was partially considered by M. Luis (1991) by providing counterexamples demonstrated that the results of Ayoub failed to capture the direct decomposition of U(R). In this article, we manage to determine the structure of U(R) when (p−1)∣k by fixing Ayoub’s approach. We also sharpen our results by introducing a system of generators for the unit group and enumerating the generators of the same order.

On [formula omitted]-cyclic codes and their applications in constructing optimal codes

Let R=F2+uF2(u2=0) and S=F2+uF2+u2F2(u3=0) be two finite commutative chain rings. This paper studies F2RS-cyclic codes, which are described as S[x]-submodules of the S[x]-module F2[x]∕〈xr−1〉×R[x]∕〈xs−1〉×S[x]∕〈xt−1〉. We study their generator polynomials and the minimal generating sets. We classify each case of the generating sets separately and determine the size of each such case. Free F2RS-cyclic codes and separable codes are discussed, and the structural properties of dual codes of free F2RS-cyclic codes are investigated. Moreover, we determine the relationship between the generator polynomials of free F2RS-cyclic codes and their duals. As applications, we provide several examples of optimal and near-optimal binary codes which are obtained from the Gray images of F2RS-cyclic codes.

Roulette games and depths of words over finite commutative rings

In this paper, we propose three new turn-based two player roulette games and provide positional winning strategies for these games in terms of depths of words over finite commutative rings with unity. We further discuss the feasibility of these winning strategies by studying depths of codewords of all repeated-root $$(\alpha +\gamma \beta )$$ -constacyclic codes of prime power lengths over a finite commutative chain ring $${\mathcal {R}},$$ where $$\alpha $$ is a non-zero element of the Teichmuller set of $${\mathcal {R}},$$ $$\gamma $$ is a generator of the maximal ideal of $${\mathcal {R}}$$ and $$\beta $$ is a unit in $${\mathcal {R}}.$$ As a consequence, we explicitly determine depth distributions of all repeated-root $$(\alpha +\gamma \beta )$$ -constacyclic codes of prime power lengths over $${\mathcal {R}}$$ .