This paper presents a new class of easily invertible cir-culant matrices, defined by exploiting the isomorphism from the ring M_n of n×n circulant matrices over GF(p) to the ring R_n=GF(p)[x]/(x~n - 1) of the polynomials modulo (x~n - 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.
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