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How to construct the generator matrix and convert it into systematic form
  1. The generator polynomial of a $(7, 4)$ cyclic code is $g(D) = 1 + D2 + D3$

    • Find all the valid code words of this code in non-systematic and in systematic form.

    • Construct the generator matrix G and convert it into systematic form.

  2. Consider the a (8, 5) cyclic code defined by the generator polynomial

    • Find the systematic cyclic code word for the input sequence m.

    • Determine the syndrome polynomial s(D) if the received code is 11010110 and verify whether or not this code is error-free.

    • Construct the generator matrix, G, in systematic form.

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