- 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.

Generator polynomial g(x)Example:

- Construction of G from g(x)
- Use g(x) to form the kth row of G.
- Use kth row to form (k-1) row by a shift left, or xg(x).
- 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).
- Continue to form rows from the row below.
- 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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