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Published online by Cambridge University Press: 15 December 2016
In this chapter, we present an algebraic construction of a special type of LDPC code whose Tanner graph has a very specific structure. For an LDPC code of this type, its Tanner graph is locally connected. Every VN is only connected to the CNs that are confined to a (small) span of ρcol consecutive locations and every CN is only connected to VNs that are confined to a (small) span of ρrow consecutive locations. We call such constraints on the connections between VNs and CNs of a Tanner graph (ρcol,ρrow)-span-constraints. With this span-constraint, the Tanner graph of such an LDPC code is actually a chain of small Tanner graphs in which each graph is connected to its adjacent graphs on either side of it, except the first and the last ones. An LDPC code of this type is called a span-constrained LDPC code. The SC-LDPC code investigated in [59, 60, 22, 86] is a type of span-constrained LDPC code.
An SC-LDPC code is an LDPC convolutional (LDPC-C) code viewed from a graphical point of view (or a spatial coupling point of view) [49, 104, 66, 67, 92]. An LDPC-C code [49, 104] is specified by a bi-infinite parity-check matrix whose nonzero entries are confined to a diagonal band of a certain width ρrow and a certain depth ρcol. The nonzero entries in every row are confined to a span of ρrow consecutive locations and the nonzero entries in every column are confined to a span of ρcol consecutive locations. With these constraints on the locations of the nonzero entries of the parity-check matrix of an LDPC-C code, every VN in its Tanner graph is only connected to the CNs that are confined to a span of ρcol consecutive locations and every CN is only connected to VNs that are confined to a span of ρrow consecutive locations. These constraints on the locations of the nonzero entries of the parity-check matrix of an LDPC-C code lead to the graphical (ρcol,ρrow)-span-constraint as mentioned above. Hence, an LDPC-C code is a span-constrained LDPC code, an SC-LDPC code.
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