Quasi-Cyclic Representation and Vector Representation of RS-LDPC Codes

RS-LDPC codes, constructed based on the codewords of Reed-Solomon (RS) codes with two information symbols, are an important class of LDPC codes. In this paper, we present two representations, namely, quasi-cyclic (QC) representation and vector representation, for RS-LDPC codes. Under the first representation, most part of the parity-check matrix of a full-length RS-LDPC code consists of circulant permutation matrices and zero matrices. As a result, the class of codes can enjoy the advantages in hardware implementation as QC-LDPC codes. In addition, the base matrix under the QC representation of an RS-LDPC code can be explicitly given such that the rank of its parity-check matrix can be analyzed combinatorially. Under the second representation, each permutation matrix in the parity-check matrix of an RS-LDPC code is defined by a nonbinary vector, whose entries are a permutation of entries in the field from which the RS code is constructed. Then, the “affine invariance” property is proved for full-length RS-LDPC codes, which can facilitate the structural analysis of the codes.