Cryptography Reference
In-Depth Information
Again from the extended Hamming code, a double serial concatenation can
be elaborated in the form of a product code (Figure 6.5(a)). In this scheme,
the redundant parts of the horizontal and vertical codewords are themselves
re-encoded by elementary codes, which produce redundancy symbols denoted
w
i,j
. One useful algebraic property of this product code is the identity of the
redundancy symbols coming from the second level of encoding, in the horizontal
and vertical directions. The MHD of the code, which has a global rate 1/4, is
again given by the patterns of errors of input weight 1 and is equal to 16, that
is, the square of the MHD of the elementary code (Figure 6.5(b)). The figure
of merit
Rd
min
=4
has therefore been greatly increased compared to parallel
concatenation. To attempt to increase the rate of this code by puncturing the
redundancy symbols while keeping a good MHD is bound to fail.
Figure 6.5 - Double serial concatenation (product code) of extended Hamming codes
(global rate: 1/4). On the right: a pattern of errors of input weight 1 and total weight
16.
In conclusion, parallel concatenation cannot be used with just any elementary
code. Today, only convolutional recursive systematic codes are used in this type
of concatenation, with 2 dimensions. Serial concatenation can offer large MHD.
The choice of codes is greater: convolutional codes, recursive or not, BCH codes
or Reed-Solomon codes. However, with the same coding rates of elementary
codes, serial concatenation has a lower global rate than parallel concatenation.
6.2
Parallel concatenation and LDPC codes
LDPC codes, which are described in Chapter 9, are codes where the lines and
columns of the parity check matrix contain few 1s. LDPC codes can be seen
as a multiple concatenation of
n
−
k
parity relations containing few variables.
