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We propose a novel low-density parity-check (LDPC) decoder design methodology by introducing a special code named diagonal-LDPC (DLDPC) code. An LPDC code, defined by a parity check matrix H, can be represented by a bipartite graph. To address the complex routing problem in the LDPC decoder implementation, a partitioned bipartite-graph code is proposed and generalized to a DLDPC code having constraint of positioning 1's near the diagonal area. This structured code simplifies the routing problem [Y. Kou et al, 2001] [R.M. Tanner et al, 2001] [H. Zhang et al, 2003] and enables cell-based highly regular fully-parallel decoder design without compromising the code performance.