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.

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.

Some results on the linear codes over the finite ring F2+v1F2+⋯+vrF2

2016 ◽  
Vol 14 (01) ◽  
pp. 1650012 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Finite Ring ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Quantum Error ◽  
Quasi Cyclic Codes

In this paper, we study the structure of cyclic, quasi-cyclic codes and their skew codes over the finite ring [Formula: see text], [Formula: see text] for [Formula: see text]. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic codes over [Formula: see text] are obtained. A necessary and sufficient condition for cyclic code over [Formula: see text] that contains its dual has been given. The parameters of quantum error correcting codes are obtained from cyclic codes over [Formula: see text].

On quantum codes via cyclic codes of arbitrary length over 𝔽4 + u𝔽4

2018 ◽  
Vol 10 (03) ◽  
pp. 1850033 ◽  
Author(s):  
Amit Sharma ◽  
Ramakrishna Bandi ◽  
Maheshanand Bhaintwal
Keyword(s):  
Linear Code ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Quantum Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arbitrary Length ◽  
Gray Image ◽  
Necessary And Sufficient

In this paper, we study cyclic codes over [Formula: see text]. A necessary and sufficient condition for a cyclic code over [Formula: see text] to contain its dual is determined. The odd and even length cases are discussed separately to obtain above condition. It is shown that Gray image of a cyclic code over [Formula: see text] containing its dual is a linear code over [Formula: see text] which also contains its dual. We have then obtained the parameters of corresponding CSS-quantum codes over [Formula: see text]. By augmentation, we construct codes with dual-containing property from codes of smaller size containing their duals. Through this construction, we have obtained some optimal quantum codes over [Formula: see text]. Some examples have been given to illustrate the results.

On skew cyclic codes over a mixed alphabet and their applications to DNA codes

2021 ◽  
pp. 2150143
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Nuh Aydin
Keyword(s):  
Finite Field ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dna Codes ◽  
Skew Cyclic Codes ◽  
Necessary And Sufficient

In this paper, we introduce skew cyclic codes over the mixed alphabet [Formula: see text], where [Formula: see text] is the finite field with 4 elements and [Formula: see text]. Our results include a description of the generator polynomials of such codes and a necessary and sufficient condition for an [Formula: see text]-skew cyclic code to be reversible complement.

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.

scholarly journals Applications of Fox's derivation in determining the generators of a group

2000 ◽  
Vol 61 (1) ◽  
pp. 27-32
Author(s):  
Wan Lin
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Quotient Group ◽  
Sufficient Conditions ◽  
Inverse Function ◽  
Sufficient Condition ◽  
Inverse Function Theorem ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Necessary And Sufficient

We give a necessary and sufficient condition for a set of elements to be a generating set of a quotient group F/N, where F is the free group of rank n and N is a normal subgroup of F. Birman's Inverse Function Theorem is a corollary of our criterion. As an application of this criterion, we give necessary and sufficient conditions for a set of elements of the Burnside group B (n,p) of exponent p and rank n to be a generating set.

scholarly journals Integral Cayley Multigraphs over Abelian and Hamiltonian Groups

10.37236/2742 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Matt DeVos ◽  
Roi Krakovski ◽  
Bojan Mohar ◽  
Azhvan Sheikh Ahmady
Keyword(s):  
Boolean Algebra ◽  
Character Sums ◽  
Sufficient Condition ◽  
Normal Subgroups ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Theoretic Proof ◽  
Necessary And Sufficient ◽  
Hamiltonian Groups ◽  
Integral Cone

It is shown that a Cayley multigraph over a group $G$ with generating multiset $S$ is integral (i.e., all of its eigenvalues are integers) if $S$ lies in the integral cone over the boolean algebra generated by the normal subgroups of $G$. The converse holds in the case when $G$ is abelian. This in particular gives an alternative, character theoretic proof of a theorem of Bridges and Mena (1982). We extend this result to provide a necessary and sufficient condition for a Cayley multigraph over a Hamiltonian group to be integral, in terms of character sums and the structure of the generating set.

scholarly journals Ideal structure of Zq + uZq and Zq + uZq-cyclic codes

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4199-4214
Author(s):  
Raj Kumar ◽  
Maheshanand Bhaintwal ◽  
Ramakrishna Bandi
Keyword(s):  
Cyclic Code ◽  
Cyclic Codes ◽  
Gray Map ◽  
Sufficient Condition ◽  
Ideal Structure ◽  
Necessary And Sufficient Condition ◽  
Complete Ideal ◽  
Necessary And Sufficient

In this paper, we study cyclic codes of length n over R = Zq + uZq, u2 = 0, where q is a power of a prime p and (n; p) = 1. We have determined the complete ideal structure of R. Using this, we have obtained the structure of cyclic codes and that of their duals through the factorization of xn-1 over R. We have also computed total number of cyclic codes of length n over R. A necessary and sufficient condition for a cyclic code over R to be self-dual is presented. We have presented a formula for the total number of self-dual cyclic codes of length n over R. A new Gray map from R to Z2rp is defined. Using Magma, some good cyclic codes of length 4 over Z9 + uZ9 are obtained.

scholarly journals Relation Between Groups with Basis Property and Groups with Exchange Property

2016 ◽  
Vol 24 (2) ◽  
pp. 5-14
Author(s):  
Al Khalaf Khalaf ◽  
Mohammed Alkadhi
Keyword(s):  
Prime Order ◽  
Basis Property ◽  
Polycyclic Group ◽  
Exchange Property ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generating Set ◽  
Necessary And Sufficient ◽  
Minimal Generating Set

AbstractA group G is called a group with basis property if there exists a basis (minimal generating set) for every subgroup H of G and every two bases are equivalent. A group G is called a group with exchange property, if x∉〈X〉 ⋀ x∈〈X∪{y}〉, then y∈〈X∪{x}〉, for all x, y ∈ G and for every subset X⊆G. In this research, we proved the following: Every polycyclic group satisfies the basis property. Every element in a group with the exchange property has a prime order. Every p-group satisfies the exchange property if and only if it is an elementary abelian p-group. Finally, we found necessary and sufficient condition for every group to satisfy the exchange property, based on a group with the basis property.

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].

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