In this paper, we construct protograph-based spatially coupled low-densityparity-check (SC-LDPC) codes by coupling together a series of L disjoint, oruncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L,we obtain a flexible family of code ensembles with varying rates and framelengths that can share the same encoding and decoding architecture forarbitrary L. We demonstrate that the resulting codes combine the best featuresof optimized irregular and regular codes in one design: capacity approachingiterative belief propagation (BP) decoding thresholds and linear growth ofminimum distance with block length. In particular, we show that, forsufficiently large L, the BP thresholds on both the binary erasure channel(BEC) and the binary-input additive white Gaussian noise channel (AWGNC)saturate to a particular value significantly better than the BP decodingthreshold and numerically indistinguishable from the optimal maximuma-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When allvariable nodes in the coupled chain have degree greater than two,asymptotically the error probability converges at least doubly exponentiallywith decoding iterations and we obtain sequences of asymptotically good LDPCcodes with fast convergence rates and BP thresholds close to the Shannon limit.Further, the gap to capacity decreases as the density of the graph increases,opening up a new way to construct capacity achieving codes on memorylessbinary-input symmetric-output (MBS) channels with low-complexity BP decoding.
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