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In this paper, we present a novel way for solving the main problem ofdesigning the capacity approaching irregular low-density parity-check (LDPC)code ensemble over binary erasure channel (BEC). The proposed method is muchsimpler, faster, accurate and practical than other methods. Our method does notuse any relaxation or any approximate solution like previous works. Our methodworks and finds optimal answer for any given check node degree distribution.The proposed method was implemented and it works well in practice withpolynomial time complexity. As a result, we represent some degree distributionsthat their rates are close to the capacity with maximum erasure probability andmaximum code rate.

Optimal Rate and Maximum Erasure Probability LDPC Codes in Binary Erasure Channel