Girth analysis of polynomial-based time-invariant LDPC convolutional codes

Low-Density Parity-Check convolutional codes (LDPCccs) can be efficiently decoded by a pipelined sub-optimal Sum Product Algorithm. The latter may suffer, however, from convergence problems, due to cycles in the Tanner graph. To improve the decoding performance, we analyze the cycle properties, based on the connections between monomials in the polynomial syndrome former (transposed parity-check matrix in polynomial form) of time-invariant LDPCccs. Due to specific structures in the polynomial syndrome former, some cycles are unavoidable no matter what monomials are placed in the polynomial syndrome former. It is shown that large-weight entries in the polynomial syndrome former lead to small girth, while monomial or empty entries, which can break short unavoidable cycles, may result in large girth. A novel algorithm is proposed to generate “good” LDPCccs with respect to their cycle properties: destructive structures in the polynomial syndrome former leading to small girth are systematically avoided.