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 3 and girth 6 is 4. In this paper, we concentrate on a well-known algebraic-based construction of girth-6 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 $3times n$ submatrices of an exponent matrix in constructing girth-6 QC-LDPC codes whose ETSs have the size of at least 5. For structures on finite field $mathbb{F}_q$, where $q$ is a power of 2, all non-isomorphic $3times n$ submatrices of the exponent matrix which yield QC-LDPC codes free of small ETSs  are presented.
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