Decoding of Error Correction Codes

• Decoding of received code words is very similar to encoding.
• Let the received binary sequence be r(1) through r(7).
• For the (7,4) code considered previously, the following must be computed
  
  $$\begin{array}{cl} s(1) & = r(1) \oplus r(4) \oplus r(5) \opl... ...) & = r(3) \oplus r(4) \oplus r(5) \oplus r(7)\\ \end{array}$$
  
  

• The quantities above are computed such that the following conditions are met:
  • For any error-free code word s(1)=s(2)=s(3)=0.
  • Example: codeword 0001 011 yields s(1)=s(2)=s(3)=0.
  • Any single error leads to a different combination of s(1), s(2), s(3).