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

840 ERROR CORRECTING CODES
Figure 14. 16
Product code
formed by two
encoders
separated by a
block interleaver.
k 2 message
blocks of
k1 bits
Block
Encoder 1 interleaver Encoder 2 Channel
k 2
codewords
of length n1
n 1 codewords
oflength k 2
n 1 codewords
oflength n 2
Message bits Code 1
Message
blocks
Code 2
with larger depths can better handle longer bursts of errors, at the cost of larger memory
and longer encoding and decoding delays. A more general interleaver can pseudorandomly
reorder the data bits inside the interleaver and output the bits in an order known to both the
transmitter and the receiver. Such an interleaver is known as a random interleaver. Random
interleavers are generally more effective in combating both random and burst errors.
Because they do not generate outputs following a fixed order, there is a much smaller probability
of receiving a burst of error bits in a codeword because of certain random error
patterns.
Product Code
Interleaved code can be generalized by further encoding the interleaved codewords. The resulting
code can be viewed as a large codeword that must satisfy two parity checks ( or constraints).
Figure 14.16 illustrates how to form a product code from two systematic block codes that are
known as component codes. The first is an (n1, k1) code and the second is an (n2, k2) code.
More specifically, a rectangular block of k1 x k2 message bits is encoded by two encoders. First,
k2 blocks of k1 message bits is encoded by the first encoder into k2 codewords of the (n 1 , k1)
code. Then an n1 x k 2 block interleaver sends n1 blocks of k2 bits into the second encoder.
The second (n2, k2) encoder adds n2 - k2 parity bits for each of the n1 blocks, generating n1
codewords of the (n2, k2) code for the channel to transmit.
The use of a product code is a simple way to combine two block codes into a single more
powerful code. In a product code, every code bit is constrained by two sets of parities, one
from each of the two codes.
Concatenated Codes
Note from the block diagram of the product code that a block interleaver connects the two
component codes. More generally, as shown in Fig. 14. 17, the two component codes need not