Improved Check Node Decomposition for Linear Programming Decoding

For the linear programming decoding (LPD) proposed by Feldman et al., the number of constraints increases exponentially with check degrees. By decomposing a high-degree check node into a number of degree-3 check nodes, the number of constraints grows linearly with check degrees. In this letter, we show that the size of the LPD can be reduced by decomposing a high-degree check node into a number of degree-4 check nodes. The LPD using the degree-4 decomposition leads to almost the same number of constraints as using the degree-3 decomposition, while the number of auxiliary variable nodes is less than half of the one using the degree-3 decomposition. Moreover, when decomposing a high degree check node into a number of check nodes with degree d, d>4, the number of constraints increases rapidly and the size of the LPD becomes larger than the degree-4 decomposition. It is demonstrated on an LDPC code and a BCH code that the decoding time of the degree-4 decomposition is the smallest among the different decomposition methods.