LDPC Code Designs, Constructions, and Unification by Li Juane & Lin Shu & Abdel-Ghaffar Khaled & Ryan William E. & Costello Jr Daniel J

Author:Li, Juane & Lin, Shu & Abdel-Ghaffar, Khaled & Ryan, William E. & Costello, Jr, Daniel J.
Language: eng
Format: epub
Publisher: Cambridge University Press
Published: 2016-11-29T16:00:00+00:00

9.2 Type-1 QC-SC-LDPC Codes

Let Hb,sp,sc(τ,m,n) be the binary CPM-dispersion of Λq,sp,sc(τ,m,n). Then, Hb,sp,sc(τ,m,n) is a semi-infinite array of binary CPMs and ZMs of size (q – 1) × (q – 1) with period τ. It follows from the 2 × 2 SM-constrained property of the array Λq,sp,sc(τ,m,n) that the array Hb,sp,sc(τ,m,n) satisfies the RC-constraint. The null space of the array Hb,sp,sc(τ,m,n) gives a periodically time-varying CPM-QC-SC-LDPC code, denoted by Cb,sp,sc(τ), with period τ. The Tanner graph, denoted by b,sp,sc(Λ), of the CPM-QC-SC-LDPC code Cb,sp,sc(τ), contains no cycle of length 4 and hence has girth at least 6. It is a periodically time-varying LDPC-C code with constraint length en(q – 1) (in symbols). The CPM-QC-SC-LDPC code Cb,sp,sc(τ) constructed based on the array Λq,sp,sc(τ,m,n) is referred to as a type-1 CPM-QC-SC-LDPC code.

For 0 ≤ j < r, let CPM(R0,j) be the binary CPM-dispersion of the m×n matrix R0,j over GF(q). From the SP-construction point of view, the parity-check matrix Hb,sp,sc(τ,m,n) of the time-varying CPM-QC-SC-LDPC code Cb,sp,sc(τ) is constructed by using the set

(9.3)

of m(q – 1) ×n(q – 1) matrices (m×n array of CPMs of size (q – 1) × (q – 1)) as the replacement set R and the binary matrix Bsp given by (9.4) as the SP-base matrix. Hence, the CPM-QC-SC-LDPC code Cb,sp,sc(τ) is also a CPM-QC-SP-LDPC code.

From the PTG-construction point of view, the Tanner graph q,sp,sc(τ,m,n) of the first τ-span subarray Bq,sp,sc(τ,m,n) of the array Λq,sp,sc(τ,m,n) may be regarded as the protograph for the PTG-construction of the CPM-QC-SC-LDPC code Cb,sp,sc(τ). The edges of q,sp,sc(τ,m,n) are labeled with nonzero elements in GF(q). In the PTG-based construction of the parity-check matrix Hb,sp,sc(τ,m,n) of the code Cb,sp,sc(τ), there are two expansions of the protograph q,sp,sc(τ,m,n), first expanding q,sp,sc(τ,m,n) into q,sp,sc(Λ) by taking an infinite number of copies of q,sp,sc(τ,m,n) and connecting them into a chain and then expanding q,sp,sc(Λ) by binary CPM-dispersing the q-ary label of each edge in q,sp,sc(Λ) to construct the Tanner graph b,sp,sc(Λ) of the code Cb,sp,sc(τ) as described above. The adjacency matrix of b,sp,sc(Λ) gives the parity-check matrix Hb,sp,sc(τ,m,n) of the code Cb,sp,sc(τ). Therefore, the CPM-QC-SC-LDPC code Cb,sp,sc(τ) can be viewed as a QC-PTG-LDPC code.

Since Hb,sp,sc(τ,m,n) is the CPM-dispersion of Λq,sp,sc(τ,m,n), the Tanner graph of the time-varying CPM-QC-SC-LDPC code Cb,sp,sc(τ) is a chain of identical subgraphs, each being an expansion of the graph q,sp,sc(τ,m,n) by a factor of q – 1. Each of these subgraphs is the Tanner graph of the CPM-dispersion of Bq,sp,sc(τ,m,n), the first τ-span subarray of the semi-infinite array Λq,sp,sc(τ,m,n).

Since a type-1 CPM-QC-SC-LDPC code is an LDPC-C code, the methods devised for decoding an LDPC-C code [74, 47] can be applied to decode the CPM-QC-SC-LDPC code Cb,sp,sc(τ). The rate of the code Cb,sp,sc(τ) is at least (n – m)∕n.

(9.4)

Example 9.1. In this example, we use the field GF(127) for constructing a CPM-QC-SC-LDPC code. First, we construct a 126 × 126 cyclic matrix Bq,sp,p over GF(127) in the form given by (8.2) which satisfies the 2×2 SM-constraint. Factor 127-1 = 126 as the product of 63 and 2, and set r = 63 and l = 2.

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