I'm looking at creating sparse generator matrices for cyclic codes of a given length and dimension. A generator matrix of an [n,k] cyclic code can be expressed as

$G = \begin{bmatrix}g_0 & g_1 & \cdots& g_{n-k-1}& 1 & 0 & \cdots & 0\\ 0 &g_0 & g_1 & \cdots& g_{n-k-1}& 1 & 0 & 0\\ \vdots & & \ddots & & & \ddots & \ddots &&\\0 & \cdots & 0& g_0 & g_1 & \cdots& g_{n-k-1}& 1\end{bmatrix}$

Where the generator polynomial of the code is $g(x) = g_0 + g_1 x + \cdots + g_{n-k-1}x^{n-k-1} + x^{n-k}$

To get some guarantees on what sparsities are possible for a given n and k, I'm looking for bounds (lower and upper) on the number of non-zero generator polynomial coefficients in the lowest weight generator polynomial for n and k.

Or relating to my original problem, are there bounds on the sparsity of $G$'s of the above form?