ℤ2(ℤ2 + uℤ2)-Additive cyclic codes and their duals

2016 ◽  
Vol 08 (02) ◽  
pp. 1650027 ◽  
Author(s):  
B Srinivasulu ◽  
Maheshanand Bhaintwal
Keyword(s):  
Structural Properties ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Free Module ◽  
Necessary Condition ◽  
Spanning Set

In this paper, we study some structural properties of [Formula: see text]-additive cyclic codes in [Formula: see text] as [Formula: see text]-submodules of [Formula: see text], where [Formula: see text]. The generators for these codes are obtained and a minimal spanning set is determined for even and odd [Formula: see text] separately with arbitrary [Formula: see text]. We also determine the generators of duals of the [Formula: see text]-additive cyclic codes for odd [Formula: see text]. A necessary condition for a 1-generator [Formula: see text]-cyclic code to be a [Formula: see text]-free module is obtained.

A note on cyclic codes over ℤ4 + uℤ4

2016 ◽  
Vol 08 (01) ◽  
pp. 1650017 ◽  
Author(s):  
Rama Krishna Bandi ◽  
Maheshanand Bhaintwal
Keyword(s):  
Local Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Ideal Structure ◽  
Complete Ideal ◽  
Spanning Set

In this paper, we have studied cyclic codes over the ring [Formula: see text], [Formula: see text]. We have provided the general form of the generators of a cyclic code over [Formula: see text] and obtained a minimal spanning set for such codes and determined their ranks. We have determined a necessary condition and a sufficient condition for cyclic codes over [Formula: see text] to be [Formula: see text]-free. For [Formula: see text], we have shown that [Formula: see text] is a local ring, and the complete ideal structure of [Formula: see text] is determined. Some examples are presented.

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].

On skew cyclic codes over a semi-local ring

2015 ◽  
Vol 07 (04) ◽  
pp. 1550042 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Local Ring ◽  
Structural Properties ◽  
Decomposition Method ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Gray Map ◽  
Gray Image ◽  
Skew Cyclic Codes

In the present paper, we study skew cyclic codes over the finite semi-local ring [Formula: see text], where [Formula: see text] and [Formula: see text] is an odd prime. We define a Gray map from [Formula: see text] to [Formula: see text] and investigate the structural properties of skew cyclic codes over [Formula: see text] using decomposition method. It is proved that the Gray image of a skew cyclic code of length [Formula: see text] over [Formula: see text] is a skew [Formula: see text]-quasi-cyclic code of length [Formula: see text] over [Formula: see text]. Further, it is shown that the skew cyclic codes over [Formula: see text] are principally generated.

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.

scholarly journals Enumeration of self-dual cyclic codes of some specific lengths over finite fields

2018 ◽  
Vol 10 (03) ◽  
pp. 1850031 ◽  
Author(s):  
Supawadee Prugsapitak ◽  
Somphong Jitman
Keyword(s):  
Characteristic Function ◽  
Finite Field ◽  
Finite Fields ◽  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Efficient Algorithms ◽  
Important Class ◽  
Field Characteristic ◽  
Positive Integers

Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length [Formula: see text] over a finite field if and only if [Formula: see text] and the field characteristic are even. The enumeration of such codes has been given under both the Euclidean and Hermitian products. However, in each case, the formula for self-dual cyclic codes of length [Formula: see text] over a finite field contains a characteristic function which is not easily computed. In this paper, we focus on more efficient ways to enumerate self-dual cyclic codes of lengths [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are positive integers. Some number theoretical tools are established. Based on these results, alternative formulas and efficient algorithms to determine the number of self-dual cyclic codes of such lengths are provided.

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.

scholarly journals The Derivation of Structural Properties of LM-g Splines by Necessary Condition of Optimal Control

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Xiongwei Liu ◽  
Xinjian Zhang ◽  
Lizhi Cheng
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Structural Properties ◽  
Optimal Control Theory ◽  
State Constraints ◽  
Necessary Conditions ◽  
Necessary Condition ◽  
Interpolating Splines ◽  
The Relationship ◽  
Optimization And Optimal Control

The structural properties of LM-g splines are investigated by optimization and optimal control theory. The continuity and structure of LM-g splines are derived by using a class of necessary conditions with state constraints of optimal control and the relationship between LM-g interpolating splines and the corresponding L-g interpolating splines. This work provides a new method for further exploration of LM-g interpolating splines and its applications in the optimal control.

ON THE ALGEBRAIC PARAMETERS OF CONVOLUTIONAL CODES WITH CYCLIC STRUCTURE

2006 ◽  
Vol 05 (01) ◽  
pp. 53-76 ◽  
Author(s):  
HEIDE GLUESING-LUERSSEN ◽  
BARBARA LANGFELD
Keyword(s):  
Polynomial Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Convolutional Codes ◽  
Complete Characterization ◽  
Field Size ◽  
Cyclic Structure ◽  
Skew Polynomial Ring ◽  
Principal Ideals ◽  
Skew Polynomial

In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [4] that only certain combinations of the algebraic parameters (field size, length, dimension, and Forney indices) can occur for such cyclic codes. We will investigate whether all these combinations can indeed be realized by a suitable cyclic code and, if so, how to construct such a code. A complete characterization and construction will be given for minimal cyclic codes. It is derived from a detailed investigation of the units in the skew-polynomial ring.

Quantum codes from cyclic codes over F3 + vF3

2014 ◽  
Vol 12 (06) ◽  
pp. 1450042 ◽  
Author(s):  
Mohammad Ashraf ◽  
Ghulam Mohammad
Keyword(s):  
Commutative Ring ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Quantum Codes ◽  
Quantum Code ◽  
Chain Ring ◽  
Arbitrary Length ◽  
Orthogonal Codes ◽  
Finite Commutative Ring ◽  
A Chain

Let R = F3 + vF3 be a finite commutative ring, where v2 = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F3 as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code.

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.

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