Coding and decoding

• If m is a message word, then its code word c is given by c = mG (40)

• HcT = 0 if and only if c is a code word (42)

• Let r be an n-tuple where r = c+e (43)

• The syndrome, s, is given by s = HrT

• s = H(c+e)T = HcT+HeT = 0+HeT = HeT (44)

• Decoding a received word r

  • Compute the syndrome of r
  • Decode the syndrome to produce e
  • Compute c = r + e

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