We show that a $Q$-polynomial association scheme with Krein array $\{m, m-1, m(r^2-1)/r^2, m-r^2+1; 1, m/r^2, r^2-1, m\}$ and $m$ odd does not exist. This Krein array is feasible for all integers $m$ and $r$ such that $0 < 2(r^2-1) \le m \le r(r-1)(r+2)$ and $m(r+1)$ is even. A scheme with such a Krein array has a (pseudo)-Latin square strongly regular graph with parameters $(v, k, \lambda, \mu) = (m^2, (m-1)r^2, m + r^2(r^2-3), r^2(r^2-1))$ as its $Q$-polynomial quotient. Therefore, we say that such a scheme is of Latin square type.
There are several examples of $Q$-polynomial association schemes with such a Krein array. For $(r, m) = (2, 6)$ and $(r, m) = (3, 16)$, this Krein array is realized by the schemes of roots of the lattices $E_6$ and OBW16. For $(r, m) = (2^{ij}, 2^{i(2j+1)})$, there are $\phi(2j+1)/2$ non-isomorphic examples arising from Kasami codes for each choice of positive integers $i$ and $j$ (here, $\phi$ represents Euler's totient function). In particular, the Krein array obtained by setting $i = j = 1$ uniquely determines the halved $8$-cube.
In the case when $r$ is a prime power and $m = r^3$, the formal dual of this parameter set (i.e., a distance-regular graph with the corresponding intersection array) is realized by a Pasechnik graph.
%display latex
import drg
Such a scheme would have $2m^2$ vertices.
m, r = var("m r")
p = drg.QPolyParameters([m, m-1, m*(r^2-1)/r^2, m-r^2+1], [1, m/r^2, r^2-1, m])
p.order(factor=True, simplify=2)
This scheme is $Q$-bipartite, so it has $q^h_{ij} = 0$ whenever $h+i+j$ is odd, or $h, i, j$ do not satisfy the triangle inequality.
p.kreinParameters(factor=True, simplify=True)
The Krein parameters and intersection numbers can be checked to be nonnegative for all integers $m$ and $r$ such that $1 < r^2 \le m \le r(r-1)(r+2)$. The intersection numbers are integral when $r$ is odd - however, when it is even, $m$ must also be even.
p.set_vars([m, r])
p.pTable(factor=True, simplify=True)
By the absolute bound for $(1, 1)$, such a scheme is only feasible when $m \ge 2(r^2-1)$.
p.check_absoluteBound(expand=True, factor=True, simplify=True)
Let us check that the smallest admissible case with $m$ odd, namely $(r, m) = (3, 17)$, satisfies the known feasibility conditions. We skip the family nonexistence check since the Krein array of the members of family with $r, m$ odd is already included.
p.subs(r == 3, m == 17).check_feasible(skip=["family"])
We now compute the triple intersection numbers with respect to three vertices $x, y, z$ such that $x$ is in relation $1$ with $y$ and $z$, and $y$ is in relation $2$ with $z$. Note that we have $p^2_{11} = r^2 (r^2-1)/2 > 0$ for all $r \ge 2$, so such triples must exist. The parameter $\alpha$ will denote the number of vertices in relation $1$ to all of $x, y, z$.
S112 = p.tripleEquations(1, 1, 2, params={'alpha': (1, 1, 1)})
S112[1, 1, 3]
We see that the above triple intersection number can only be integral when $m$ is even. We thus conclude that a $Q$-polynomial association scheme with Krein array $\{m, m-1, m(r^2-1)/r^2, m-r^2+1; 1, m/r^2, r^2-1, m\}$ and $m$ odd does not exist.