This question is related to Greg Slodkowicz's question and the discussion about this paper.
We define a de Bruijn sequence of order $k$ over the set of $l$ distinct symbols as a cyclic sequence of length $l^k$, containing all possible letters sub-sequences of length $k$ exactly once. In the mentioned paper, they say that for any arbitrary Galois field - $GF(p^n)$, for $p$ a prime number and $n$ an integer, a de Bruijn sequence of order $k$ over all elements can be generated by the following recursion:
\begin{equation} s_i=\theta_{k-1}s_{i-1} + \dots + \theta_0s_{i-k}, \end{equation}
with arithmetic performed over $GF(p^n)$ and $\theta_i\in GF(p^n)$ so the polynomial $\sum_{i=0}^{k-1}\theta_i x^i$ is primitive. In Greg Slodkowicz's question, it was mentioned that the considered polynomial seems to be actually $x^k - \sum_{i=0}^{k-1}\theta_i x^i$. I find it hard to get the right primitive polynomial by which in the paper they illustrated the construction of the de Bruijn sequence over $GF(4)$ with what seems to be $k=2$ in Figure 3a. If we have $GF(4)=\{0,1,2,3\}$ with the known addition and multiplication:
addition and multiplication tables,
how does one get, using primitive polynomial and the suggested recursion equation, the generated sequence in the paper in Figure 3a? i.e.,
\begin{equation} 011210331302232, \end{equation}
so, with the addition of a single $0$ to the beginning, we get the de Bruijn sequence over $GF(4)$ with k=2.