A class of invertible circulant matrices for QC-LDPC codes

Abstract

This paper presents a new class of easily invertible circulant matrices, defined by exploiting the isomorphism from the ring Mn of n times n circulant matrices over GF(p) to the ring Rn = GF(p)[x]/(xn - 1) of the polynomials modulo (xn - 1). Such class contains matrices free of 4-length cycles that, if sparse, can be included in the parity check matrix of QC-LDPC codes. Bounds for the weight of their inverses are also determined, that are useful for designing sparse generator matrices for these error correcting codes.