49)
The number of code words, Ncw, in a sphere of radius, t, is given as
(5.50)
Given the earlier example of the (7,4) code, we know there are 27 possible code words of which only
24 are used. Hence the number of code words in a sphere of radius t 1 is given as Ncw 8.
In an effort to bring this subsection on block codes to some closure, we will briefly present an
overview of the block codes discussed. We started by simply providing generator matrices and polynomials
for certain block codes. It is out of the scope of this book to discuss how they were obtained.
Suffice it to say, these encoding rules or mapping were taken from classical reference books and journal
papers. We presented some examples of Hamming codes, BCH codes, and cyclic codes. Syndrome
Ncw at
i0 an
i b
Pb
1
n an
it1
i # an
i b # pi # (1 p)ni
Pm an
mt1an
mb # pm # (1 p)nm
Pm an
mt1
P(m,n)
P(m,n) an
mb # pm # (1 p)nm
PERFORMANCE IMPROVEMENT TECHNIQUES 241
t
FIGURE 5.21 Error correcting pictorial view.
decoding principles were presented for the FEC decoder operations. Lastly, an example of a very
powerful technique called erasure decoding was presented.
5.1.3 Convolutional Codes
In this subsection we will discuss an FEC encoding technique using convolutional codes [10].
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