The Feng-Rao Designed Minimum Distance of Binary Linear Codes and Cyclic Codes

2005 ◽  
pp. 167-182
Author(s):  
Junru Zheng ◽  
Takayasu Kaida ◽  
Kyoki Imamura
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Binary Linear Codes

scholarly journals Results on Binary Linear Codes With Minimum Distance 8 and 10

2011 ◽  
Vol 57 (9) ◽  
pp. 6089-6093 ◽  
Author(s):  
Iliya Georgiev Bouyukliev ◽  
Erik Jacobsson
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Binary Linear Codes

scholarly journals Maximal Arcs, Codes, and New Links Between Projective Planes of Order 16

10.37236/9008 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Mustafa Gezek ◽  
Rudi Mathon ◽  
Vladimir D. Tonchev
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Projective Planes ◽  
Upper And Lower Bounds ◽  
Dual Codes ◽  
Binary Linear Codes ◽  
Incidence Matrices ◽  
Maximal Arc ◽  
Maximal Arcs ◽  
Even Order

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The  binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.

scholarly journals Isodual and self-dual codes from graphs

2021 ◽  
Vol 32 (1) ◽  
pp. 49-64
Author(s):  
S. Mallik ◽  
◽  
B. Yildiz ◽  
Keyword(s):  
Minimum Distance ◽  
Linear Codes ◽  
Type I ◽  
Type Ii ◽  
Dual Codes ◽  
Generator Matrix ◽  
Combinatorial Interpretation ◽  
Binary Linear Codes ◽  
Graph Theoretic ◽  
Graph Classes

Binary linear codes are constructed from graphs, in particular, by the generator matrix [In|A] where A is the adjacency matrix of a graph on n vertices. A combinatorial interpretation of the minimum distance of such codes is given. We also present graph theoretic conditions for such linear codes to be Type I and Type II self-dual. Several examples of binary linear codes produced by well-known graph classes are given.

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.

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