- Generator Polynomial: The generator polynomial ( g(x) ) is used to multiply the message polynomial. For your case, the generator polynomial is given as ( g(x) = x^2 + \alpha^4 x + \alpha^3 ).
- Message Polynomial: The message polynomial is represented by the vector [0 0 0 1 0], which corresponds to ( x^3 ).
- Encoding: The encoding process involves multiplying the message polynomial by ( x^{n-k} ) and then dividing by the generator polynomial ( g(x) ). The remainder of this division is then added to the original message polynomial multiplied by ( x^{n-k} ).

- Message Vector: The original message vector is [0 0 0 1 0], which corresponds to ( x^3 ).
- Multiply by (x^{n-k}): Since ( n-k = 2 ), this means shifting the message polynomial by 2 degrees, resulting in ( x^5 ).
- Polynomial Division: The shifted message polynomial is divided by the generator polynomial (g(x) ). The remainder of this division is the error correction code.
- Encoded Message: The encoded message is the original message polynomial (shifted) plus the error correction code.