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Iterative decoding of low-density parity check codes (LDPC) using the message-passing algorithm have proved to be extraordinarily effective compared to conventional maximum-likelihood decoding. However, the lack of any structural regularity in these essentially random codes is a major challenge for building a practical low-power LDPC decoder. In this paper, we jointly design the code and the decoder to induce the structural regularity needed for a reduced-complexity parallel decoder architecture. This interconnect-driven code design approach eliminates the need for a complex interconnection network while still retaining the algorithmic performance promised by random codes. Moreover, we propose a new approach for computing reliability metrics based on the BCJR algorithm that reduces the message switching activity in the decoder compared to existing approaches. Simulations show that the proposed approach results in power savings of up to 85.64% over conventional implementations.