The primitive idempotents of irreducible constacyclic codes and LCD cyclic codes

2019 ◽  
Vol 12 (1) ◽  
pp. 29-52
Author(s):  
Zexia Shi ◽  
Fang-Wei Fu
Keyword(s):  
Cyclic Codes ◽  
Constacyclic Codes ◽  
Primitive Idempotents

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.

scholarly journals Primitive Idempotents of Irreducible Cyclic Codes of Length n

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yuqian Lin ◽  
Qin Yue ◽  
Yansheng Wu
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Matrix Method ◽  
Cyclic Codes ◽  
Prime Divisors ◽  
Primitive Idempotents ◽  
Irreducible Cyclic Codes

Let Fq be a finite field with q elements and n a positive integer. In this paper, we use matrix method to give all primitive idempotents of irreducible cyclic codes of length n, whose prime divisors divide q-1.

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.

EXPLICIT DETERMINATION OF CERTAIN MINIMAL ABELIAN CODES AND THEIR MINIMUM DISTANCES

2012 ◽  
Vol 05 (01) ◽  
pp. 1250002
Author(s):  
Pooja Grover ◽  
Ashwani K. Bhandari
Keyword(s):  
Abelian Groups ◽  
Cyclic Codes ◽  
Group Algebras ◽  
Primitive Idempotents ◽  
Abelian Codes ◽  
Complete Set ◽  
Minimum Distances ◽  
Minimal Codes ◽  
Explicit Determination

In this paper minimal codes for several classes of non-cyclic abelian groups have been constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Some classes of non-p-groups have also been considered. The minimum distances of such abelian codes have been discussed and compared to the minimum distances of cyclic codes of same lengths and dimensions over the same field.

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>.

Generalized cyclotomic numbers and cyclic codes of prime power length over Z4

2019 ◽  
Vol 12 (05) ◽  
pp. 1950085 ◽  
Author(s):  
Sudhir Batra ◽  
Sonal Jain
Keyword(s):  
Prime Power ◽  
Cyclic Codes ◽  
Self Duality ◽  
Primitive Idempotents ◽  
Multiplicative Order ◽  
Cyclotomic Numbers ◽  
Explicit Expressions

Generalized cyclotomic numbers of order [Formula: see text] with respect to an odd prime power are obtained. Hence, explicit expressions for primitive idempotents in the ring [Formula: see text] are obtained in two cases, when the multiplicative order of 2 modulo [Formula: see text] is [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime. Orthogonality and self-duality of some [Formula: see text] cyclic codes are also discussed. Further, a method for obtaining cyclic self-dual/isodual codes of length [Formula: see text] over [Formula: see text] is given.

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