Paper Info
Title
On generator and parity-check polynomial matrices of generalized quasi-cyclic codes
Abstract
Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular generator polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of generator polynomial matrices of GQC codes, including a relation formula between generator polynomial matrices and parity-check polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Gröbner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all generator polynomial matrices with given cyclic intervals.
Year
DOI
Venue
2015
10.1016/j.ffa.2015.02.003
Finite Fields and Their Applications
Keywords
Field
DocType
gr&#246,05e20,11t71,zout's identity,automorphism group,buchberger's algorithm,b&#233,error-correcting codes,94b05,extended euclidean algorithm,circulant matrix,bner basis,bezout s identity,grobner basis,buchberger s algorithm
Discrete mathematics,Combinatorics,Group code,Polynomial,Algebra,Matrix (mathematics),Block code,Polynomial code,Burst error-correcting code,Reed–Solomon error correction,Linear code,Mathematics
Journal
Volume
Issue
ISSN
34
C
1071-5797
Citations 
PageRank 
References 
0
0.34
17
Authors
1
Name
Order
Citations
PageRank
Hajime Matsui1188.14