An algebraic construction of QC-LDPC codes based on powers of primitive elements in a finite field and free of small ETSs

An (a,b) elementary trapping set (ETS), where a and b denote the size and the number of unsatisfied check nodes in the ETS, influences  the performance of  low-density parity-check (LDPC) codes.  The smallest size of an ETS in LDPC codes with column weight ۳ and girth ۶ is ۴. In this paper, we concentrate on a well-known algebraic-based construction of girth-۶ QC-LDPC codes based on powers of a primitive element in a finite field \mathbb{F}_q. For this structure, we provide the sufficient conditions to obtain ۳\times n submatrices of an exponent matrix in constructing girth-۶ QC-LDPC codes whose ETSs have the size of at least ۵. For structures on finite field \mathbb{F}_q, where q is a power of ۲, all non-isomorphic ۳\times n submatrices of the exponent matrix which yield QC-LDPC codes free of small ETSs  are presented.

Keywords:

QC-LDPC codes , girth , Tanner graph , elementary trapping set