Let
m
≥
3 be an integer and let
t
be a positive integer (i.e. the error correction capability
of the code). Then, binary BCH codes exist for (
n, k
) where
n
= 2
m
−
1 ;
n
−
mt
≤
k < n
.
(74)
Generator polynomials for a large set of binary BCH codes can be found in a number of
digital communication texts (e.g. Proakis [2]).
Example 7.14:
In Table 7.10-1 of Proakis [2], find the generator polynomial
for the (15
,
7) binary BCH code. What is the codeword for information vector
X
m
= [0001111].
From Proakis [2], Table 7.10-1, the generator polynomial coefficients are given
by the octal number
7 2 1 = [1 1 1 0 1 0 0 0 1]
.
The polynomial has degree
n
−
k
= 8. It is
g
(
p
) =
p
8
+
p
7
+
p
6
+
p
4
+ 1
.
[1 1 1 0 1 0 0 0 1]
(75)
[1 1 1 1 0 0 0]
C
m
= [0 0 0
,
1 0 1 1
,
1 0 1 1 1
,
1 1 1]
Kevin Buckley - 2010
140
7.5.9
Other Linear Block Codes
In this Subsection, we overviewed a number of important binary linear block codes. These
are important in their own right, and they are also often used as building blocks for other
codes, such as concatenated and product codes, which we will investigate later. Descriptions
of other useful codes, such as Euclidean geometry, projective geometry, quadratic residue
and fire codes, can be found in Lin & Costello’s text [1].
We have considered binary linear block codes, there performance parameters
d
min
and the
weighting function
wn
j
, and some encoder structures. We now turn to decoding algorithms
for and performance analysis of these codes.
Kevin Buckley - 2010
141
7.6
Binary Linear Block Code Decoding & Performance Analysis
In a digital communication system, we receive noisy symbols.
For a system employing
block coding and a given modulation scheme, we will receive a sequence of waveforms, that
represent a block codeword
C
m
, which is corrupted by noise. Under the assumptions that:
•
the sequence of information bits is statistically independent,
•
a memoryless modulation scheme is employed,
•
channel is memoryless; and
•
the noise is white,
the optimum receiver front end will consist of a filter matched to the symbol waveform
followed by a symbol-rate sampler. This receiver structure is illustrated in Figure 60. The
matched filter output vector
r
contains the samples corresponding to a transmitted codeword
C
m
. That is, it contains the
n
matched filter output samples corresponding to the
n
symbols
representing
C
m
. Under the assumptions stated above, this vector can be processed (without
any information of the received data for other codewords) to optimally estimate
C
m
. The
exact statistical characteristics
r
, and this the communications system performance, will
depend on the modulation scheme employed.
linear binary
block code
encoder
modulator
C
m
X
m
communication
channel
quantizer
decoder
ML
r
X
m
^
C
m
^
X
m
^
C
m
^
X
m
^
C
m
^
soft decision
decoder
decisions
hard
hard decision
decoder
n T
c
R
Y
matched filter
r (t)
r (t)
Figure 60: Digital communication system with block channel encoding.
