Syndromes

• The combination of s(1), s(2), s(3) is called the syndrome of the code.
• The following table summarizes syndromes associated with the most likely error patterns:
  

  Error pattern	syndrome
  0 0 0 0 0 0 0	0 0 0
  0 0 0 0 0 0 1	0 1 1
  0 0 0 0 0 1 0	1 1 0
  0 0 0 0 1 0 0	1 1 1
  0 0 0 1 0 0 0	1 0 1
  0 0 1 0 0 0 0	0 0 1
  0 1 0 0 0 0 0	0 1 0
  1 0 0 0 0 0 0	1 0 0

• Now, to correct an error the following procedure can be used:
  • Compute the syndrome.
  • Look up the error pattern associated with the syndrome.
  • Form the exclusive or of received word and error pattern.
• With this procedure, any single bit error can be corrected.
• Example: Assume 0000 011 was received.
• The syndrom for this word is 101.
• From the table above, the error pattern is 0001 000.
• 0000 011 $\oplus$ 0001 000 = 0001 011.
• Notice, how the error pattern indicates the position of the bit in error.
• Remark: The feedback shift register circuit below can be used for computing the syndrome.
• The shift registers are initially zero.
• Received bits are shifted into the shift register one bit at a time.
• After 7 shift, the contents of the register equals the syndrome.

  

Figure: Feedback shift register circuit for computing syndrome for rate $\frac{7}{4}$ block code.