The application of linear complexity of sequences to lower bounds on the minimum distance of cyclic codes

1990 ◽  
Vol 7 (4) ◽  
pp. 312-316
Author(s):  
Li Yuanxing ◽  
Liang Chuanjia
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Linear Complexity ◽  
Cyclic Codes

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.

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.

Use of Grobner bases to decode binary cyclic codes up to the true minimum distance

1994 ◽  
Vol 40 (5) ◽  
pp. 1654-1661 ◽  
Author(s):  
Xuemin Chen ◽  
I.S. Reed ◽  
T. Helleseth ◽  
T.K. Truong
Keyword(s):  
Minimum Distance ◽  
Cyclic Codes ◽  
Gröbner Bases ◽  
Grobner Bases

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

On some bounds on the minimum distance of cyclic codes over finite fields

2014 ◽  
Vol 76 (2) ◽  
pp. 173-178
Author(s):  
Ferruh Özbudak ◽  
Seher Tutdere ◽  
Oğuz Yayla
Keyword(s):  
Finite Fields ◽  
Minimum Distance ◽  
Cyclic Codes

Double Circulant Self-Dual and LCD Codes Over ℤp2

2019 ◽  
Vol 30 (03) ◽  
pp. 407-416
Author(s):  
Daitao Huang ◽  
Minjia Shi ◽  
Patrick Solé
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Exact Enumeration ◽  
Lcd Codes ◽  
Gray Maps ◽  
Double Circulant Codes ◽  
Circulant Codes

We study double circulant LCD codes over [Formula: see text] for all odd primes [Formula: see text] and self-dual double circulant codes over [Formula: see text] for primes [Formula: see text]. We derive exact enumeration formulae, and asymptotic lower bounds on the minimum distance of the [Formula: see text]-ary images of these codes by the classical Gray maps.

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