On quasi-cyclic codes over $${\mathbb{Z}_q}$$

Bhaintwal, Maheshanand; Wasan, Siri Krishan

doi:10.1007/s00200-009-0110-8

Cited by 39 publications

(22 citation statements)

References 28 publications

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AbstractBy applying the theory of linear algebra, one can obtain some properties of these codes which generalize several well known results for free cyclic codes represented in [1,4].

Mathematics Subject Classification: 94B05, 11T71, 47A15Keywords: finite chain rings, free cyclic codes, invariant submodules

Preliminary NotesLet R be a finite chain ring and R n the R-module of rank n consisting of row vectors over R. We writen is called a cyclic code of length n if (a n−1 , a 0 , . .

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confidence: 53%

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Free cyclic codes as invariant submodules over finite chain rings

Wu¹

2013

imf

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By applying the theory of linear algebra, one can obtain some properties of these codes which generalize several well known results for free cyclic codes represented in [1,4]. Mathematics Subject Classification: 94B05, 11T71, 47A15Keywords: finite chain rings, free cyclic codes, invariant submodules Preliminary NotesLet R be a finite chain ring and R n the R-module of rank n consisting of row vectors over R. We writen is called a cyclic code of length n if (a n−1 , a 0 , . . . , a n−2 ) ∈ C, for all (a 0 , a 1 , . . . , a n−1 ) ∈ C., where I n−1 is the identity matrix of order n − 1, and denote by ε i the vector in R n with zeros every where but in position i that holds, i = 0, 1, . . . , n − 1. Then {ε 0 , ε 1 , . . . , ε n−1 } is the standard R-basis of R n . We denote by T the set of R-linear transformation of R n . Then T is an R-algebra with operations of linear transformations and scalar multiplications with elements of R.

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“…

AbstractBy applying the theory of linear algebra, one can obtain some properties of these codes which generalize several well known results for free cyclic codes represented in [1,4].

Mathematics Subject Classification: 94B05, 11T71, 47A15Keywords: finite chain rings, free cyclic codes, invariant submodules

Preliminary NotesLet R be a finite chain ring and R n the R-module of rank n consisting of row vectors over R. We writen is called a cyclic code of length n if (a n−1 , a 0 , . .

…”

supporting

confidence: 53%

“…By applying the theory of linear algebra, one can obtain some properties of these codes which generalize several well known results for free cyclic codes represented in [1,4].…”

Section: Introductionmentioning

confidence: 55%

Free cyclic codes as invariant submodules over finite chain rings

Wu¹

2013

imf

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“…As described in [13] and in [3], we can also view a skew QC code C of length s and index over G R(q, m) as a left R s -submodule of G R(q,m )[x,σ ] x s −1 by identifying R s with G R(q, m ) through the one-to-one correspondence…”

Section: Theorem 5 a Module σ -Cyclic Code Of Length N Over G R(q M)mentioning

confidence: 98%

“…Bhaintwal (B) Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India e-mail: mahesfma@iitr.ernet.in the commutative setting of finite fields or that of finite chain-rings is well studied [2,3,15]. In [1], Abualrub et al have studied skew quasi-cyclic codes over finite fields as a generalization of classical QC codes in the new setting of a skew polynomial rings.…”

Section: Introductionmentioning

confidence: 98%

Skew quasi-cyclic codes over Galois rings

Bhaintwal

2011

Des. Codes Cryptogr.

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Recently there has been a lot of interest on algebraic codes in the setting of skew polynomial rings. In this paper we have studied skew quasi-cyclic (QC) codes over Galois rings. We have given a necessary and sufficient condition for skew cyclic codes over Galois rings to be free, and determined a distance bound for free skew cyclic codes. A sufficient condition for 1-generator skew QC codes to be free is determined. Some distance bounds for free 1-generator skew QC codes are discussed. A canonical decomposition of skew QC codes is presented.

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“…Since the revelation in 1994 [17], there has been a lot of interest in codes over finite rings. The structure of a certain type of codes over many rings are determined such as negacyclic, cyclic, quasi-cyclic, consta cyclic codes in [6,11,20,21,22,23,26,32]. Many methods and many approaches are applied to produce certain types of codes with good parameters and properties.…”

Section: Introductionmentioning

confidence: 99%

On the Codes over a Semilocal Finite Ring

Dertli¹,

Çengellenmiş²,

Eren³

2015

ijacsa

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Abstract-In this paper, we study the structure of cyclic, quasi cyclic, constacyclic codes and their skew codes over the finite ring R. The Gray images of cyclic, quasi cyclic, skew cyclic, skew quasi cyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The parameters of quantum error correcting codes are obtained from both cyclic and negacyclic codes over R. Some examples are given. Firstly, quasi constacyclic and skew quasi constacyclic codes are introduced. By giving two inner product, it is investigated their duality. A sufficient condition for 1 generator skew quasi constacyclic codes to be free is determined.

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On quasi-cyclic codes over $${\mathbb{Z}_q}$$

Cited by 39 publications

References 28 publications

Free cyclic codes as invariant submodules over finite chain rings

Free cyclic codes as invariant submodules over finite chain rings

Skew quasi-cyclic codes over Galois rings

On the Codes over a Semilocal Finite Ring

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