On the consecutive sets of defining sequence for lower bounds on cyclic codes

2017 ◽  
Author(s):  
Junru Zheng ◽  
Takayasu Kaida
Keyword(s):  
Lower Bounds ◽  
Cyclic Codes ◽  
Defining Sequence

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.

Exact lower bounds on the codelength of three-step permutation-decodable cyclic codes

1992 ◽  
Vol 38 (6) ◽  
pp. 1812-1817 ◽  
Author(s):  
M. Jia ◽  
A. Benyamin-Seeyar ◽  
T. Le-Ngoc
Keyword(s):  
Lower Bounds ◽  
Cyclic Codes

scholarly journals The Cardinality of the Set of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

2017 ◽  
Vol 9 (2) ◽  
pp. 56
Author(s):  
Yasanthi Kottegoda ◽  
Robert Fitzgerald
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Upper Bound ◽  
Cyclic Codes ◽  
Quadratic Residue ◽  
Upper And Lower Bounds ◽  
Linear Recurring Sequences ◽  
Irreducible Cyclic Codes ◽  
Homogeneous Linear

Consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on the irreducible characteristic polynomial of degree $d$ and order $m$. We give upper and lower bounds, and in some cases the exact values of the cardinality of the set of zeros of the sequences within its least period. We also prove that the cyclotomy bound introduced here is the best upper bound as it is reached in infinitely many cases. In addition, the exact number of occurrences of zeros is determined using the correlation with irreducible cyclic codes when $(q^{d}-1)/ m$ follows the quadratic residue conditions and also when it has the form $q^{2a}-q^{a}+1$ where $a\in \mathbb{N}$.

scholarly journals Cyclic Codes via the General Two-Prime Generalized Cyclotomic Sequence of Order Two

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Xia Zhou
Keyword(s):  
Prime Number ◽  
Lower Bounds ◽  
Cyclic Codes ◽  
Prime Numbers ◽  
Binary Sequences ◽  
Generalized Cyclotomy ◽  
Minimal Polynomials ◽  
Generalized Cyclotomic Sequence ◽  
Minimum Distances

Suppose that p and q are two distinct odd prime numbers with n = p q . In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤ n was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over F l was presented and their minimal polynomials and linear complexities were derived, where l = r s with a prime number r and gcd l , n = 1 . The results have indicated that the linear complexities of these sequences are high without any special requirements on the prime numbers. Furthermore, we employed these sequences to obtain a few cyclic codes over F l with length n and developed the lower bounds of the minimum distances of many cyclic codes. It is important to stress that some cyclic codes in this paper are optimal.

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