A SPECIAL CLASS OF QUASI-CYCLIC CODES

2017 ◽  
Vol 96 (3) ◽  
pp. 513-518 ◽  
Author(s):  
MINJIA SHI ◽  
JIE TANG ◽  
MAORONG GE ◽  
LIN SOK ◽  
PATRICK SOLÉ
Keyword(s):  
Special Class ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Dual Code ◽  
Extension Field ◽  
Dual Codes ◽  
Lcd Codes ◽  
Quasi Cyclic Codes

We study a special class of quasi-cyclic codes, obtained from a cyclic code over an extension field of the alphabet field by taking its image on a basis. When the basis is equal to its dual, the dual code admits the same construction. We give some examples of self-dual codes and LCD codes obtained in this way.

scholarly journals Constacyclic and Linear Complementary Dual Codes Over Fq + uFq

2020 ◽  
Vol 70 (6) ◽  
pp. 626-632
Author(s):  
Om Prakash ◽  
Shikha Yadav ◽  
Ram Krishna Verma
Keyword(s):  
Cyclic Codes ◽  
Gray Map ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Gray Image ◽  
New Class ◽  
Sharing Scheme ◽  
Lcd Codes ◽  
Reversible Cyclic Codes

This article discusses linear complementary dual (LCD) codes over ℜ = Fq+uFq(u2=1) where q is a power of an odd prime p. Authors come up with a new Gray map from ℜn to F2nq and define a new class of codes obtained as the gray image of constacyclic codes over .ℜ Further, we extend the study over Euclidean and Hermitian LCD codes and establish a relation between reversible cyclic codes and Euclidean LCD cyclic codes over ℜ. Finally, an application of LCD codes in multisecret sharing scheme is given.

Triple Cyclic Codes Over 𝔽q + u𝔽q

2021 ◽  
pp. 1-21
Author(s):  
Ting Yao ◽  
Shixin Zhu ◽  
Binbin Pang
Keyword(s):  
Special Class ◽  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Cyclic Shift ◽  
Generating Sets ◽  
Optimal Linear ◽  
Number Formula ◽  
The Relationship ◽  
Optimal Linear Codes

Let [Formula: see text], where [Formula: see text] is a power of a prime number [Formula: see text] and [Formula: see text]. A triple cyclic code of length [Formula: see text] over [Formula: see text] is a set that can be partitioned into three parts that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as [Formula: see text]-submodules of [Formula: see text]. In this paper, we study the generator polynomials and the minimum generating sets of this kind of codes. Some optimal or almost optimal linear codes are obtained from this family of codes. We present the relationship between the generators of triple cyclic codes and their duals. As a special class of triple cyclic codes, separable codes over [Formula: see text] are discussed briefly in the end.

Abelian group factorization from cyclic and quasi-cyclic codes

2020 ◽  
pp. 2150016
Author(s):  
Abdulla Eid ◽  
Sameh Ezzat
Keyword(s):  
Abelian Group ◽  
Cyclic Code ◽  
Abelian Groups ◽  
Cyclic Codes ◽  
The Other ◽  
Algebraic Structures ◽  
Algorithmic Techniques ◽  
Zero Vector ◽  
Quasi Cyclic Codes

In this paper, we use the algebraic structures of cyclic codes and algorithmic techniques to find factorizations of abelian groups from cyclic codes. We construct specific subclasses of quasi-cyclic codes and provide the conditions with which we obtain a normalized factorization of certain abelian groups. The factorization, in both cases, is constituted by two sets, one corresponding to the cyclic code and the other corresponding to the words that represent all possible error polynomials of the cyclic code besides the zero vector.

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].

scholarly journals On Self-Dual Four Circulant Codes

2018 ◽  
Vol 29 (07) ◽  
pp. 1143-1150 ◽  
Author(s):  
Minjia Shi ◽  
Hongwei Zhu ◽  
Liqin Qian ◽  
Patrick Solé
Keyword(s):  
Special Class ◽  
Cyclic Codes ◽  
Primitive Root ◽  
Index Formula ◽  
Generator Matrix ◽  
Artin Primitive Root Conjecture ◽  
Circulant Codes ◽  
Quasi Cyclic Codes

Four circulant codes form a special class of [Formula: see text]-generator, index [Formula: see text], quasi-cyclic codes. Under some conditions on their generator matrix they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an infinite subclass of these codes satisfying a modified Gilbert–Varshamov bound.

scholarly journals An explicit construction of quantum codes from one-generator generalized quasi-cyclic codes

2021 ◽  
Vol 336 ◽  
pp. 04001
Author(s):  
Yu Yao ◽  
Yuena Ma ◽  
Husheng Li ◽  
Jingjie Lv
Keyword(s):  
Cyclic Codes ◽  
Explicit Construction ◽  
Error Correcting Codes ◽  
Inner Product ◽  
Quantum Codes ◽  
Dual Codes ◽  
Index 2 ◽  
Quantum Error ◽  
Hermitian Inner Product ◽  
Quasi Cyclic Codes

In this paper, we take advantage of a class of one-generator generalized quasi-cyclic (GQC) codes of index 2 to construct quantum error-correcting codes. By studying the form of Hermitian dual codes and their algebraic structure, we propose a sufficient condition for self-orthogonality of GQC codes with Hermitian inner product. By comparison, the quantum codes we constructed have better parameters than known codes.

A class of constacyclic codes over the ring Z4[u,v]/ and their Gray images

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2237-2248 ◽  
Author(s):  
Habibul Islam ◽  
Om Prakash
Keyword(s):  
Cyclic Codes ◽  
Constacyclic Codes ◽  
Gray Maps ◽  
Permutation Equivalent ◽  
Quasi Cyclic Codes

In this paper, we study (1 + 2u + 2v)-constacyclic and skew (1 + 2u + 2v)-constacyclic codes over the ring Z4 + uZ4 + vZ4 + uvZ4 where u2 = v2 = 0,uv = vu. We define some new Gray maps and show that the Gray images of (1 + 2u + 2v)-constacyclic and skew (1 + 2u + 2v)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z4. Further, we determine the structure of (1 + 2u + 2v)-constacyclic codes of odd length n.

Quasi-cyclic codes over the ring 𝔽p[u]/〈u2 − u〉

2015 ◽  
Vol 08 (04) ◽  
pp. 1550085
Author(s):  
Sukhamoy Pattanayak ◽  
Abhay Kumar Singh
Keyword(s):  
Structural Properties ◽  
Cyclic Codes ◽  
Natural Generalization ◽  
Spanning Set ◽  
The One ◽  
Quasi Cyclic Codes

Quasi-cyclic (QC) codes are a natural generalization of cyclic codes. In this paper, we study some structural properties of QC codes over [Formula: see text], where [Formula: see text] is a prime and [Formula: see text]. By exploring their structure, we determine the one generator QC codes over [Formula: see text] and the size by giving a minimal spanning set. We discuss some examples of QC codes of various length over [Formula: see text].

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