Unavoidable Cycles in Polynomial-Based Time-Invariant LDPC Convolutional Codes

Low-Density Parity-Check convolutional codes (LDPCccs) are very interesting for practical error-correction coding in wireless transmission as they have excellent performance and at the same time they allow for variable block size with low complexity encoding and decoding. As for all LDPC codes that are decoded by the sub-optimal (but highly efficient) Sum Product Algorithm, the cycles in the code graph are very important for the practical performance of the coding scheme. Time-invariant LDPCccs can be defined by a polynomial syndrome former (transposed parity-check matrix in polynomial form), that can be derived from corresponding Quasi-Cyclic (QC) LDPC block codes. Given the polynomial syndrome former with certain structures, unavoidable cycles with lengths ranging from 6 to 12 will be shown to exist. We provide some rules for designing good codes with respect to the shortest cycle in the code-graph, the girth of the code, which is a crucial parameter for its decoding performance.