39)
In the above derivation, the last line is obtained using the orthogonality constraint given earlier. Here
we see the syndrome is dependent on the error vector.
We can now outline the procedure for correcting the single errors using the syndrome decoding
method, it is as follows:
Step 1: Given calculate the syndrome, S.
Step 2: Find the column in H (or the row in HT) whose value matches the syndrome. This determines
the position of the error.
Step 3: Generate this error vector, and add it to the received code word.
(5.40)
Note the transmitted code word is extracted only if the locally generated error vector, , matches
the channel generated error vector, .
In Fig. 5.18, we provide a generalized block diagram of the error correction method based on syndrome
calculation for the (7,4) code. The received code word enters the syndrome calculation block.
The syndrome then enters the error pattern detector, which generates the local error vector. Lastly,
errors are corrected by the modulo 2 sum of the error vector and the received code word.
e S
e^
r S
e ^
C S
r S
e ^
C S
e S
e
^
r S
C S
e S
e^
r S
S e S
# HT
S x S
# G # HT e S
# HT
238 CHAPTER FIVE
r0 r1 r2 r3 r4 r5 r6
Syndrome Calculation
Error Pattern Detection
+
S1 S2 S3
+ + + + + +
Corrected Code Word
e0
r0
C0
e6
r6
C6
FIGURE 5.
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