An Exhaustive Tree Search for Stopping Sets of LDPC Codes

The indicative performance of an LDPC code may be determined from exhaustive analysis of the low-weight spectral terms of the code’s stopping sets which by definition includes the low-weight codewords. In a landmark paper, Rosnes and Ytrehus showed that exhaustive, low-weight stopping set analysis of codes whose parity-check matrix is sparse can be carried out using a bounded tree search over the length of the code. For an (n, k) code, the potential total search space is of size \(2^n\) but a good choice of bound dramatically reduces this search space to a practical size. Indeed, the choice of bound is critical to the success of the algorithm. It is shown in this chapter that an improved algorithm can be obtained if the bounded tree search is applied to a set of k information bits since the potential total search space is reduced to size \(2^k\). Since such a restriction will only find codewords and not all stopping sets, a class of bits is defined as unsolved parity bits, and these are also searched as appended bits in order to find all low-weight stopping sets. Weight spectrum and stopping set results are presented for commonly used WiMax LDPC codes in addition to some other well known LDPC codes.