scholarly journals On Automorphism Groups of Finite Chain Rings

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 681
Author(s):  
Sami Alabiad ◽  
Yousef Alkhamees
Keyword(s):  
Automorphism Group ◽  
General Condition ◽  
Automorphism Groups ◽  
Finite Ring ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Finite Chain Rings ◽  
A Chain

A finite ring with an identity is a chain ring if its lattice of left ideals forms a unique chain. Let R be a finite chain ring with invaraints p,n,r,k,k′,m. If n=1, the automorphism group Aut(R) of R is known. The main purpose of this article is to study the structure of Aut(R) when n>1. First, we prove that Aut(R) is determined by the automorphism group of a certain commutative chain subring. Then we use this fact to find the automorphism group of R when p∤k. In addition, Aut(R) is investigated under a more general condition; that is, R is very pure and p need not divide k. Based on the j-diagram introduced by Ayoub, we were able to give the automorphism group in terms of a particular group of matrices. The structure of the automorphism group of a finite chain ring depends essentially on its invaraints and the associated j-diagram.

A class of linear codes of length 2 over finite chain rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050103 ◽  
Author(s):  
Yonglin Cao ◽  
Yuan Cao ◽  
Hai Q. Dinh ◽  
Fang-Wei Fu ◽  
Jian Gao ◽  
...  
Keyword(s):  
Positive Integer ◽  
Linear Codes ◽  
Invertible Element ◽  
Irreducible Polynomial ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Finite Chain Rings ◽  
Positive Integers

Let [Formula: see text] be a finite field of cardinality [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] be positive integers satisfying [Formula: see text], and denote [Formula: see text], where [Formula: see text] is an irreducible polynomial in [Formula: see text]. In this note, for any fixed invertible element [Formula: see text], we present all distinct linear codes [Formula: see text] over [Formula: see text] of length [Formula: see text] satisfying the condition: [Formula: see text] for all [Formula: see text]. This conclusion can be used to determine the structure of [Formula: see text]-constacyclic codes over the finite chain ring [Formula: see text] of length [Formula: see text] for any positive integer [Formula: see text] satisfying [Formula: see text].

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

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 307
Author(s):  
Sami Alabiad ◽  
Yousef Alkhamees
Keyword(s):  
Unit Group ◽  
Galois Ring ◽  
Direct Decomposition ◽  
Finite Ring ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Group Of Units ◽  
Lattice Of Ideals

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.

APPLYING BUCHBERGER'S CRITERIA FOR COMPUTING GRÖBNER BASES OVER FINITE-CHAIN RINGS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350034 ◽  
Author(s):  
AMIR HASHEMI ◽  
PARISA ALVANDI
Keyword(s):  
Cyclic Codes ◽  
Gröbner Bases ◽  
Special Kind ◽  
Grobner Bases ◽  
Discrete Mathematics ◽  
Galois Rings ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Finite Chain Rings

Norton and Sălăgean [Strong Gröbner bases and cyclic codes over a finite-chain ring, in Proc. Workshop on Coding and Cryptography, Paris, Electronic Notes in Discrete Mathematics, Vol. 6 (Elsevier Science, 2001), pp. 391–401] have presented an algorithm for computing Gröbner bases over finite-chain rings. Byrne and Fitzpatrick [Gröbner bases over Galois rings with an application to decoding alternant codes, J. Symbolic Comput.31 (2001) 565–584] have simultaneously proposed a similar algorithm for computing Gröbner bases over Galois rings (a special kind of finite-chain rings). However, they have not incorporated Buchberger's criteria into their algorithms to avoid unnecessary reductions. In this paper, we propose the adapted version of these criteria for polynomials over finite-chain rings and we show how to apply them on Norton–Sălăgean algorithm. The described algorithm has been implemented in Maple and experimented with a number of examples for the Galois rings.

On repeated-root multivariable codes over a finite chain ring

2007 ◽  
Vol 45 (2) ◽  
pp. 219-227 ◽  
Author(s):  
E. Martínez-Moro ◽  
I. F. Rúa
Keyword(s):  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring

scholarly journals Communication Over Finite-Chain-Ring Matrix Channels

2014 ◽  
Vol 60 (10) ◽  
pp. 5899-5917 ◽  
Author(s):  
Chen Feng ◽  
Roberto W. Nobrega ◽  
Frank R. Kschischang ◽  
Danilo Silva
Keyword(s):  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring

scholarly journals Constacyclic Codes over Finite Chain Rings of Characteristic p

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 303
Author(s):  
Sami Alabiad ◽  
Yousef Alkhamees
Keyword(s):  
Local Rings ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Dual Codes ◽  
Chain Ring ◽  
Characteristic P ◽  
Arbitrary Length ◽  
Cyclotomic Cosets ◽  
Finite Chain Rings ◽  
Polynomial Representations

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]/<xN−λ> 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.

CAYLEY GRAPHS OVER A FINITE CHAIN RING AND GCD-GRAPHS

2016 ◽  
Vol 93 (3) ◽  
pp. 353-363 ◽  
Author(s):  
BORWORN SUNTORNPOCH ◽  
YOTSANAN MEEMARK
Keyword(s):  
Graph Theory ◽  
Prime Power ◽  
Quotient Ring ◽  
Spectral Graph Theory ◽  
Prime Power Order ◽  
Finite Chain ◽  
Circulant Graphs ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Spectral Graph

We extend spectral graph theory from the integral circulant graphs with prime power order to a Cayley graph over a finite chain ring and determine the spectrum and energy of such graphs. Moreover, we apply the results to obtain the energy of some gcd-graphs on a quotient ring of a unique factorisation domain.

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