Abstract: It is well known that performance of low-density parity check (LDPC) codes under iterative decoding is determined by the size of the smallest stopping sets in the Tanner graph. To solve this problem, we propose necessary and sufficient conditions of quasi-cyclic LDPC codes without small stopping sets. According to the proposed theorems and corollaries, we can design good QC-LDPC codes without small stopping sets which outperforms random LDPC codes by SNR=0.3 dB at BER of 1e-5 , and count the number of the small stopping sets. The method can be used effectively to evaluate performance of LDPC codes according to their stopping sets distributions. It can find the number of cycles of LDPC codes which is less complex than existing algorithms.

Key words: quasi-cyclic low density parity check (LDPC) codes, cycle
, circulant matrices, stopping sets, stopping distance, girth