Cyclic Codes

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Cyclic Codes

• Any end-around shift of a code word will produce
  another code word
• Characterized by its generator polynomial G(X)
• G(X) is a polynomial of degree (n k) or
  greater, where n is the number of bits contained
  in the complete word and k is the number of bits
  in the original information
• For binary cyclic codes the coefficients of the
  polynomial are either 0 or 1
• A cyclic code with a generator polynomial of
  degree (n - k) is called an (n, k) cyclic code
• Can detect all single errors and all multiple,
  adjacent errors affecting fewer than (n - k)
  bits.

2
Cyclic Codes

• Suppose that code word
  . It corresponds to the polynomial V(X) where
• Each n-bit code word is represented by a
  polynomial of degree (n 1) or less.
• The polynomial V(X) is called the code polynomial
  of the code word v

3
Cyclic Codes

• The code polynomials for a nonseparable cyclic
  code are generated by multiplying a polynomial
  representing the data by another polynomial known
  as the generator polynomial
• The code polynomial is generated by multiplying
  the data polynomial by the generator polynomial
  and adding the coefficients in the modulo-2
  fashion
• Example Generator polynomial,
  and need to encode the binary
  data 1101
• Data polynomial
  and V(X) D(X) G(X)
• v (10100001)

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Cyclic Codes
5
Cyclic Codes