(5.9)
Which can be rewritten as follows
(5.10)
where P is the parity matrix of size k (n k) and I is the identity matrix of size k k.
We have a resulting systematic code word because the generator matrix generates a linear block
code where the last k bits of each code word are identical to the information bits to be transmitted
(e.g., the FEC encoder input) and the remaining (nk) bits of each code word represent a linear combination
of the k information bits at the input.
BCH, Hamming, and Cyclic Codes. Bose-Chadhuri-Hocquenghem (BCH) codes are extensions to
the Hamming codes, and because of this reason we wish to first discuss Hamming codes [5].
Hamming codes are characterized by the following FEC encoder constraints
(5.11)
These codes have a minimum distance of 3 and are capable of correcting all single errors or detecting
2 or fewer errors within the block of data. Please recall our previous definitions.
(5.12)
(5.13) e dmin 1 2 errors
t jdmin 1
2 k 1 error
(m 2, 3, c) (n, k) (2m 1, 2m 1 m)
G [P Ik]
G ?‰?p11
p21
(
pk1
p12
p22
c
c
f
pk(nk)
1 0 0
c
0 1 0
c
( f
0
c 1 ??
G ?‰?g11 g12 c g1n
g21 g22 c g2n
( f
gk1 c gkn ??
C S
x S
# G
C S
Cc1, c2,c, cn D
x S
Cx1, x2,c, xkD
e dmin 1
232 CHAPTER FIVE
Since this is a single-error correcting code, syndrome decoding can be used rather nicely to perform
the error correction [6].
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