Efficient and exact evaluation of the weight spectral shape and typical minimum distance of protograph LDPC codes

In this letter, an analytical method for evaluating the weight spectral shape of a nonbinary protograph low-density parity-check code ensemble is provided. This method, which requires simultaneous solution of a system of equations, is more efficient to implement than the existing methods that require the numerical solution of a high-dimensional maximization problem. It is also shown that by adding a single equation to the proposed system, it is possible to directly solve for the typical minimum distance of the ensemble. This makes the proposed method particularly useful for the ensemble design scenario where a constraint is placed on the typical minimum distance. The proposed method is verified through the evaluation of the weight spectral shape of some example protograph ensembles.