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.

scholarly journals Strong Gröbner bases for polynomials over a principal ideal ring

2001 ◽  
Vol 64 (3) ◽  
pp. 505-528 ◽  
Author(s):  
Graham H. Norton ◽  
Ana Sǎlǎgean
Keyword(s):  
Principal Ideal ◽  
Gröbner Bases ◽  
Galois Ring ◽  
Grobner Bases ◽  
Principal Ideal Domain ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Principal Ideal Ring ◽  
Ideal Ring

Gröbner bases have been generalised to polynomials over a commutative ring A in several ways. Here we focus on strong Gröbner bases, also known as D-bases. Several authors have shown that strong Gröbner bases can be effectively constructed over a principal ideal domain. We show that this extends to any principal ideal ring. We characterise Gröbner bases and strong Gröbner bases when A is a principal ideal ring. We also give algorithms for computing Gröbner bases and strong Gröbner bases which generalise known algorithms to principal ideal rings. In particular, we give an algorithm for computing a strong Gröbner basis over a finite-chain ring, for example a Galois ring.

Cyclic self-orthogonal codes over finite chain ring

2018 ◽  
Vol 11 (06) ◽  
pp. 1850078 ◽  
Author(s):  
Abhay Kumar Singh ◽  
Narendra Kumar ◽  
Kar Ping Shum
Keyword(s):  
Prime Number ◽  
Cyclic Codes ◽  
Sufficient Condition ◽  
Galois Rings ◽  
Finite Chain ◽  
Necessary And Sufficient Condition ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Orthogonal Codes ◽  
Necessary And Sufficient

In this paper, we study the cyclic self-orthogonal codes over a finite commutative chain ring [Formula: see text], where [Formula: see text] is a prime number. A generating polynomial of cyclic self-orthogonal codes over [Formula: see text] is obtained. We also provide a necessary and sufficient condition for the existence of nontrivial self-orthogonal codes over [Formula: see text]. Finally, we determine the number of the above codes with length [Formula: see text] over [Formula: see text] for any [Formula: see text]. The results are given by Zhe-Xian Wan on cyclic codes over Galois rings in [Z. Wan, Cyclic codes over Galois rings, Algebra Colloq. 6 (1999) 291–304] are extended and strengthened to cyclic self-orthogonal codes over [Formula: see text].

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].

Quantum codes over a class of finite chain ring

2016 ◽  
pp. 39-49
Author(s):  
Mustafa Sari ◽  
Irfan Siap
Keyword(s):  
Finite Field ◽  
Cyclic Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Quantum Error

In this study, we introduce a new Gray map which preserves the orthogonality from the chain ring F_2 [u] / (u^s ) to F^s_2 where F_2 is the finite field with two elements. We also give a condition of the existence for cyclic codes of odd length containing its dual over the ring F_2 [u] / (u^s ) . By taking advantage of this Gray map and the structure of the ring, we obtain two classes of binary quantum error correcting (QEC) codes and we finally illustrate our results by presenting some examples with good parameters.

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.

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