B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

14. 1 2 Low-Density Parity Check (LDPC) Codes 855
Fi g
ure 14.22
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rate 1 /2 Tu rbo
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the theoretical
limit.
(Reproduced
with copyright
permission from
IEEE from
Ref. 14.)
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Theoretical
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means of LDPC decoding have been topics of intensive research in the coding community.
A large number of LDPC codes have been proposed as strong competitors to turbo codes,
often achieving better performance with comparable code lengths, code rates, and decoding
complexity.
LDPC codes are linear block codes with sparse parity check matrices. In essence, the parity
check matrix H consists of mostly Os and very few ls, forming a low-density parity check
matrix. LDPC codes are typically quite long (normally longer than 1000 bits) and noncyclic.
Thus, an exact implementation of the BCJR MAP decoding algorithm is quite complex and
mostly impractical. Fortunately, there are several well-established methods for decoding LDPC
codes that can achieve near-optimum performance.
The design of LDPC code is equivalent to the design of a sparse parity matrix H. Once H
has been defined, the LDPC code is the null-space of the parity matrix H. The number of ls
in the ith row of H is known as the row weight Pi, whereas the number of ls in the jth column