Threshold upper bounds and optimized design of protograph LDPC codes for the Binary Erasure Channel

Exact density evolution of protograph Low Density Parity Check (LDPC) codes over the Binary Erasure Channel (BEC) is considered. Upper bounds on the threshold are derived and expressed as single-variable minimizations. A simplified version of the upper bound is expressed in closed form in terms of the degrees of the nodes in the protograph. By maximizing the upper bound, useful conditions are derived for optimizing protographs to get thresholds close to capacity bounds. Using these conditions, a randomized construction method for good small-sized protographs is presented.