Stopping-set enumerator approximations for finite-length protograph LDPC codes

Asymptotic analysis of low-density parity-check (LDPC) code weight and stopping-set enumerators, for codewords and stopping sets which grow linearly with codelength, has aided in designing codes with linear minimum distance and low error floors. However, the analysis cannot capture the behavior of codewords and stopping sets that grow sublinearly with codelength. Thus, it is unclear how well the analysis describes behavior for finite codelengths, particularly for short codes. In this paper, we provide another perspective on protograph-based and standard LDPC ensemble enumerators, based on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, protograph structure, and precoding. Using approximations to the stopping-set enumerators, we show that for stopping sets that grow at most logarithmically with codelength, the enumerators follow a polynomial relationship with codelength, unlike the exponential relationship for linearly-growing stopping sets. Further, we analyze for what stopping-set sizes and codelengths the approximations apply.