Generator Polynomial \(g(D)\) is equivalent to the Generator Matrix in Linear Block Coding sense. Its degree is equal to the number of parity bits in the code which is \(n-k\). It is unique as it is the only code word polynomial of minimum degree \(n-k\), and any multiplication done on it is a #Code Word Polynomial.
Its property can be stated mathematically as below:
$$ \begin{align} g(D) &= 1 + \sum^{n-k-1}_{i=1} g_i D^i + D^{n-k}\\ x(D) &= a(D) g(D) \mod (D^n + 1) \end{align} $$
Where:
- \(D\) is an arbitrary real variable (see #Code Word Polynomial)
- \(x(D)\) is #the cyclic code word polynomial
- \(a(D)\) is a polynomial in \(D\)