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 NOTE ON k-GALOIS LCD CODES OVER THE RING

2020 ◽  
pp. 1-8
Author(s):  
RONGSHENG WU ◽  
MINJIA SHI
Keyword(s):  
Prime Number ◽  
Linear Code ◽  
Sufficient Condition ◽  
Finite Chain ◽  
Generator Matrix ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Lcd Codes ◽  
Lcd Code

Abstract We study the k-Galois linear complementary dual (LCD) codes over the finite chain ring $R=\mathbb {F}_q+u\mathbb {F}_q$ with $u^2=0$ , where $q=p^e$ and p is a prime number. We give a sufficient condition on the generator matrix for the existence of k-Galois LCD codes over R. Finally, we show that a linear code over R (for $k=0, q> 3$ ) is equivalent to a Euclidean LCD code, and a linear code over R (for $0<k<e$ , $(p^{e-k}+1)\mid (p^e-1)$ and ${(p^e-1)}/{(p^{e-k}+1)}>1$ ) is equivalent to a k-Galois LCD code.

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.

Skew quasi cyclic codes over 𝔽q + v𝔽q

2019 ◽  
Vol 18 (04) ◽  
pp. 1950077 ◽  
Author(s):  
Mehmet Özen ◽  
N. Tuğba Özzaim ◽  
Halit İnce
Keyword(s):  
Cyclic Codes ◽  
Quantum Codes ◽  
Computer Search ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Skew Cyclic Codes ◽  
Necessary And Sufficient ◽  
Quantum Parameters ◽  
Quasi Cyclic Codes

In this work, skew quasi cyclic codes over [Formula: see text], where [Formula: see text] are considered. The generating set for one generator skew quasi cyclic codes over [Formula: see text] is also determined. We discuss a sufficient condition for one generator skew quasi cyclic codes to be free. Furthermore, a BCH type bound is given for free one generator skew quasi cyclic codes. We investigate the dual of skew quasi cyclic codes over [Formula: see text]. We give a necessary and sufficient condition for skew cyclic codes over [Formula: see text] to contain its dual. Moreover, we construct quantum codes from skew cyclic codes over [Formula: see text]. By using computer search we give some examples about skew quasi cyclic codes and list some quantum parameters in the table.

Some results on cyclic codes over ℤq + uℤq

2015 ◽  
Vol 07 (04) ◽  
pp. 1550058 ◽  
Author(s):  
Jian Gao ◽  
Fang-Wei Fu ◽  
Ling Xiao ◽  
Rama Krishna Bandi
Keyword(s):  
Hamming Distance ◽  
Cyclic Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generating Sets ◽  
Minimum Hamming Distance ◽  
Necessary And Sufficient

Let [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, minimum generating sets of cyclic codes over [Formula: see text] are given. A necessary and sufficient condition for cyclic codes over [Formula: see text] to be [Formula: see text]-free is obtained and a BCH-type bound on the minimum Hamming distance for them is also given.

On quantum codes obtained from cyclic codes over A2

2015 ◽  
Vol 13 (03) ◽  
pp. 1550031 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Length ◽  
Necessary And Sufficient ◽  
Quantum Error

In this paper, quantum codes from cyclic codes over A2 = F2 + uF2 + vF2 + uvF2, u2 = u, v2 = v, uv = vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A2, then so is Ψ(C), where Ψ is a Gray map. A necessary and sufficient condition for cyclic codes over A2 that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A2.

scholarly journals Bell Numbers Modulo a Prime Number, Traces and Trinomials

10.37236/3532 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Luis H. Gallardo ◽  
Olivier Rahavandrainy
Keyword(s):  
Prime Number ◽  
Finite Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bell Numbers ◽  
Fixed Integer ◽  
Bell Number ◽  
Modulo P ◽  
Necessary And Sufficient

Given a prime number $p$, we deduce from a formula of Barsky and Benzaghou and from a result of Coulter and Henderson on trinomials over finite fields, a simple necessary and sufficient condition $\beta(n) =k\beta(0)$ in $\mathbb{F}_{p^p}$ in order to resolve the congruence $B(n) \equiv k \pmod{p}$, where $B(n)$ is the $n$-th Bell number, and $k$ is any fixed integer. Several applications of the formula and of the condition are included, in particular we give equivalent forms of the conjecture of Kurepa that $B(p-1)$ is $\neq 1$ modulo $p$.

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.

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