Cyclic codes andλ1+λ2u+λ3v+λ4uv-constacyclic codes overFp+uFp+vFp+uvFp

2017 ◽  
Vol 306 ◽  
pp. 86-91 ◽  
Author(s):  
Xiying Zheng ◽  
Bo Kong
Keyword(s):  
Cyclic Codes ◽  
Constacyclic 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.

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.

A Note on Skew Cyclic Codes over a Class of Rings

2020 ◽  
Vol 27 (04) ◽  
pp. 703-712
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta
Keyword(s):  
Finite Field ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
General Setting ◽  
Constacyclic Codes ◽  
Arbitrary Length ◽  
Skew Cyclic Codes

We study skew cyclic codes over a class of rings [Formula: see text], where each [Formula: see text] [Formula: see text] is a finite field. We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or a quasi-cyclic code over R. Moreover, we discuss possible extension of our results in the more general setting of [Formula: see text]-dual skew constacyclic codes over R, where δR is an automorphism of R.

A class of constacyclic codes from group algebras

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2917-2923
Author(s):  
Mehmet Koroglu ◽  
Irfan Siap
Keyword(s):  
Cyclic Codes ◽  
Group Algebras ◽  
Constacyclic Codes ◽  
Algebraic Structures ◽  
Engineering Applications ◽  
Negacyclic Codes ◽  
Zero Divisors ◽  
First Time

Constacyclic codes are preferred in engineering applications due to their efficient encoding process via shift registers. The class of constacyclic codes contains cyclic and negacyclic codes. The relation and presentation of cyclic codes as group algebras has been considered. Here for the first time, we establish a relation between constacyclic codes and group algebras and study their algebraic structures. Further, we give a method for constructing constacyclic codes by using zero-divisors in group algebras. Some good parameters for constacyclic codes which are derived from the proposed construction are also listed.

On Cyclic and Negacyclic Codes of Length 8ps Over Fpm + uFpm

2020 ◽  
Vol 87 (3-4) ◽  
pp. 231
Author(s):  
Saroj Rani
Keyword(s):  
Positive Integer ◽  
Algebraic Structure ◽  
Cyclic Codes ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Negacyclic Codes

In this paper, we establish the algebraic structure of all cyclic and negacyclic codes of length 8<em>p</em><sup>s</sup> over the chain ring Fp<sup>m</sup> + uFp<sup>m</sup> in terms of their generator polynomials, where u<sup>2</sup> = 0 and s is a positive integer and p is an odd prime. We also find out the number of codewords in each of these cyclic codes. Besides this, we determine duals of cyclic codes and list self-dual cyclic and negacyclic codes of length 8<em>p</em><sup>s</sup> over Fp<sup>m</sup> + uFp<sup>m</sup>. Also, we determine μ and -constacyclic codes of length 8<em>p</em><sup>s</sup> over Fp<sup>m</sup> + uFp<sup>m</sup>.

A study of constacyclic codes over the ring ℤ4[u]/〈u2 − 3〉

2018 ◽  
Vol 10 (04) ◽  
pp. 1850056
Author(s):  
Tushar Bag ◽  
Habibul Islam ◽  
Om Prakash ◽  
Ashish K. Upadhyay
Keyword(s):  
Cyclic Codes ◽  
Constacyclic Codes ◽  
Gray Maps ◽  
Permutation Equivalent ◽  
Quasi Cyclic Codes

In this paper, we study [Formula: see text]-constacyclic codes over the ring [Formula: see text], where [Formula: see text] for [Formula: see text] and [Formula: see text], respectively. We define some new Gray maps from [Formula: see text] to the copies of [Formula: see text]. It is shown that Gray images of [Formula: see text]-constacyclic codes over [Formula: see text] are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over [Formula: see text]. Further, we extend and obtain cyclic codes, [Formula: see text]-constacyclic codes and permutation equivalent to quasi-cyclic codes over [Formula: see text], respectively, as Gray images of skew [Formula: see text]-constacyclic codes over [Formula: see text].

Optimal b-symbol constacyclic codes with respect to the Singleton bound

2019 ◽  
Vol 19 (08) ◽  
pp. 2050151 ◽  
Author(s):  
Hai Q. Dinh ◽  
Xiaoqiang Wang ◽  
Jirakom Sirisrisakulchai
Keyword(s):  
Finite Field ◽  
Cyclic Codes ◽  
Maximum Distance ◽  
Constacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable

Let [Formula: see text] be the finite field of order [Formula: see text], where [Formula: see text] is a power of odd prime [Formula: see text]. Assume that [Formula: see text], [Formula: see text] are nonzero elements of the finite field [Formula: see text] such that [Formula: see text]. In this paper, we determine the [Formula: see text]-distance of [Formula: see text]-constacyclic codes with generator polynomials [Formula: see text] of length [Formula: see text], where [Formula: see text] and [Formula: see text]. As an application, all maximum distance separable (MDS) [Formula: see text]-symbol constacyclic codes of length [Formula: see text] over [Formula: see text] are established. Among other results, we construct several classes of new MDS symbol-pair codes with minimum symbol-pair distance six or seven by using repeated-root cyclic codes of length [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is an odd prime.

On the linear codes over the ring Rp

2016 ◽  
Vol 08 (02) ◽  
pp. 1650036 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Prime Number ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Constacyclic Codes ◽  
Nontrivial Automorphism ◽  
Macwilliams Identities ◽  
Skew Cyclic Codes ◽  
Quantum Error ◽  
Quasi Cyclic Codes

Some results are generalized on linear codes over [Formula: see text] in [15] to the ring [Formula: see text], where [Formula: see text] is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over [Formula: see text] are obtained. The parameters of quantum error correcting codes are obtained from negacyclic codes over [Formula: see text]. A nontrivial automorphism [Formula: see text] on the ring [Formula: see text] is determined. By using this, the skew cyclic, skew quasi-cyclic, skew constacyclic codes over [Formula: see text] are introduced. The number of distinct skew cyclic codes over [Formula: see text] is given. The Gray images of skew codes over [Formula: see text] are obtained. The quasi-constacyclic and skew quasi-constacyclic codes over [Formula: see text] are introduced. MacWilliams identities of linear codes over [Formula: see text] are given.

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