Cycle analysis of time-invariant LDPC convolutional codes

Time-invariant low-density parity-check convolutional codes (LDPCccs) can be constructed from a polynomial form of a parity-check matrix that defines quasi-cyclic LDPC block codes based on circulant matrices. Based on this polynomial matrix, we discuss the relationships between the polynomial domain and the time domain parity-check and syndrome former matrices with respect to cycle properties. We present a novel, simple way to describe cycles in the polynomial version of the syndrome former matrix and we exploit this concept in a new cycle counting algorithm.