Binary Cyclic Codes

• A subclass of linear block codes are the binary cyclic codes.

• If ck is a code word in a binary cyclic code then a lateral (or cyclic) shift of it is also a code word.

• If

  

  is the code word then

  

  and

  

  can also be code words.

• Advantages:
  • Ease of syndrome calculation
  • Simple and efficient coding/decoding structure

• Cyclic codes restrict the form of the G matrix

• The cyclic nature of these codes indicates that there is a underlying pattern.

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  Generator polynomial g(x)

  

  Example:

  

• Construction of G from g(x)
  1. Use g(x) to form the kth row of G.
  2. Use kth row to form (k-1) row by a shift left, or xg(x).
  3. If (k-1) row in not in standard form then add kth row to shifted row, i.e.., (k-1) row becomes xg(x) + g(x).
  4. Continue to form rows from the row below.
  5. If the (k-j) row not in standard form the add g(x) to the shifted row.

  

  Codes for this example are:

  

• Standard generator polynomials

• Performance of standard codes
  • Detects all single errors
  • Detects all double errors
  • Detects all odd number of errored bits
  • Detects all burst errors of 16 or fewer bits
  • Detects 99.997% of all burst of 17 or fewer bits

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