  Strongly walk regular graphs (SWRGs or $s$-SWRGs) form a natural
generalization of strongly regular graphs (SRGs) where paths of length~2 are
replaced by paths of length~$s$. They can be constructed as coset graphs of the
duals of projective three-weight codes whose weights satisfy a certain
equation. We provide classifications of the feasible parameters of these codes
in the binary and ternary case for medium size code lengths. For the binary
case, the divisibility of the weights of these codes is investigated and
several general results are shown. It is known that an $s$-SWRG has at most 4
distinct eigenvalues $k > \theta_1 > \theta_2 > \theta_3$, and that the triple
$(\theta_1, \theta_2, \theta_3)$ satisfies a certain homogeneous polynomial
equation of degree $s - 2$ (Van Dam, Omidi, 2013). This equation defines a
plane algebraic curve; we use methods from algorithmic arithmetic geometry to
show that for $s = 5$ and $s = 7$, there are only the obvious solutions, and we
conjecture this to remain true for all (odd) $s \ge 9$.
