Reed Solomon Code Working in GF(32). Polynomial is $x^5+x^2+1$. $\alpha$ is primitive element. $t = 3$. RS code. $n = 31$, $k = 25$. I have obtained generator polynomial $x^6+\alpha^{10}x^5+\alpha^{30}x^4+\dots$
How do I obtain generator matrix? I believe I write down coefficients in increasing order of $x$ over $31$ columns and $25$ rows and keep shifting since it is cyclic. Is this correct?

Yes you're absolutely correct. If, say, $G(x) = g_0+g_1x^1+\dots+g_{n-k}x^{n-k}$, then the Generator matrix (a possible one; certainly not systematic. For a systematic generator matrix, please look at the other answer from Prof. Sarwate which is much better) is given as: $$G=\left[ \begin{array}{} g_0 & g_1 & g_2 & \dots & g_{n-k} & 0 & \dots & 0 & 0 & 0\\ 0 & g_0 & g_1 & \dots & g_{n-k-1} & g_{n-k} & 0 &\dots & 0 & 0 \\ 0 & 0 & g_0& \dots& g_{n-k-2}& g_{n-k-1}& g_{n-k}& 0& \dots& 0\\. & & & & & & & & & . \\ 0&0&0&\dots & g_0 & g_1 & g_2 &\dots & \dots & g_{n-k}\end{array}\right]$$
