QUASI-CYCLIC CODES FROM CYCLIC-STRUCTURED DESIGNS WITH GOOD PROPERTIES

2011 ◽  
Vol 03 (02) ◽  
pp. 223-243
Author(s):  
CHRISTOS KOUKOUVINOS ◽  
DIMITRIS E. SIMOS
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Optimization Problem ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Mathematical Structure ◽  
Combinatorial Optimization Problem ◽  
Supersaturated Designs ◽  
Statistical Designs ◽  
Quasi Cyclic Codes

In this paper, one-generator binary quasi-cyclic (QC) codes are explored by statistical tools derived from design of experiments. A connection between a structured cyclic class of statistical designs, k-circulant supersaturated designs and QC codes is given. The mathematical structure of the later codes is explored and a link between complementary dual binary QC codes and E(s2)-optimal k-circulant supersaturated designs is established. Moreover, binary QC codes of rate 1/3, 1/4, 1/5, 1/6 and 1/7 are found by utilizing a genetic algorithm. Our approach is based on a search for good or best codes that attain the current best-known lower bounds on the minimum distance of linear codes, formulated as a combinatorial optimization problem. Surveying previous results, it is shown, that our codes reach the current best-known lower bounds on the minimum distance of linear codes with the same parameters.

scholarly journals CONSTRUCTIONS OF SUBSYSTEM CODES OVER FINITE FIELDS

2009 ◽  
Vol 07 (05) ◽  
pp. 891-912 ◽  
Author(s):  
SALAH A. ALY ◽  
ANDREAS KLAPPENECKER
Keyword(s):  
Quantum Information ◽  
Lower Bounds ◽  
Minimum Distance ◽  
Physical State ◽  
Cyclic Codes ◽  
Upper And Lower Bounds ◽  
Error Protection ◽  
Tensor Factor ◽  
Code Constructions ◽  
Quantum Error

Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. They generalize all major quantum error protection schemes, and therefore are especially versatile. This paper introduces numerous constructions of subsystem codes. It is shown how one can derive subsystem codes from classical cyclic codes. Methods to trade the dimensions of subsystem and co-subsystem are introduced that maintain or improve the minimum distance. As a consequence, many optimal subsystem codes are obtained. Furthermore, it is shown how given subsystem codes can be extended, shortened, or combined to yield new subsystem codes. These subsystem code constructions are used to derive tables of upper and lower bounds on the subsystem code parameters.

scholarly journals q-ary Quasi-Cyclic Codes on Matroid Theory

2018 ◽  
Vol 173 ◽  
pp. 03095
Author(s):  
Li Xiuli ◽  
Zhao Pengcheng ◽  
Zhou Dangsheng
Keyword(s):  
Minimum Distance ◽  
Cyclic Codes ◽  
Matroid Theory ◽  
Generator Matrix ◽  
The Relationship ◽  
Quasi Cyclic Codes

In this paper, rate 1/p q-ary systematic quasi-cyclic codes are constructed based on matroid theory. The relationship between the generator matrix and minimum distance d is derived. Examples and algorithm are presented.

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].

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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