Let $Y$ be a tight $4$-design in the Hamming scheme $H(n, q)$ (i.e., an orthogonal array of strength $4$ meeting the Rao bound). Noda proved that then one of the following holds:
Cases 1 and 2 uniquely determine a binary code with $4$ data bits and one parity bit (i.e., the dual of the binary repetition code of length $5$), and the dual of ternary Golay code, respectively. No examples are known for Case 3.
Gavrilyuk, Suda and Vidali show that a $Q$-polynomial association scheme with $d = 4$ classes and Krein array $\{9a^2 - 4, 9a^2 - 9, 10, 1; 1, 2, 9a^2 - 9, 9a^2 - 4\}$ would arise from an example of Case 3. We may write $r = 3a$, thus obtaining the Krein array $\{r^2 - 4, r^2 - 9, 10, 1; 1, 2, r^2 - 9, r^2 - 4\}$. Here, we show that no corresponding schemes exist, and consequently also no example of Case 3 exists. Note however that we consider all positive integral values of $r$.
import drg
A scheme with the above Krein array would have $r^2 (r^2 - 1)/2$ vertices.
r, t = var("r t")
p = drg.QPolyParameters([r^2 - 4, r^2 - 9, 10, 1], [1, 2, r^2 - 9, r^2 - 4])
p.order(factor = True, simplify = True)
1/2*(r + 1)*(r - 1)*r^2
Such a scheme is $Q$-antipodal, so it has $q^h_{ij} = 0$ whenever $h, i, j$ do not satisfy the triangle inequality, or $h+i+j > 2d$ and $d-h, d-i, d-j$ do not satisfy the triangle inequality.
p.kreinParameters(factor = True, simplify = True)
0: [ 1 0 0 0 0] [ 0 (r + 2)*(r - 2) 0 0 0] [ 0 0 1/2*(r + 3)*(r + 2)*(r - 2)*(r - 3) 0 0] [ 0 0 0 5*(r + 2)*(r - 2) 0] [ 0 0 0 0 5] 1: [ 0 1 0 0 0] [ 1 4 (r + 3)*(r - 3) 0 0] [ 0 (r + 3)*(r - 3) 1/2*(r + 4)*(r + 3)*(r - 3)*(r - 4) 5*(r + 3)*(r - 3) 0] [ 0 0 5*(r + 3)*(r - 3) 20 5] [ 0 0 5*r^6/((r^2 - 4)*(r^2 - 9)) - 5*r^4/(r^2 - 4) - 170*r^4/((r^2 - 4)*(r^2 - 9)) + 125*r^2/(r^2 - 4) + 1845*r^2/((r^2 - 4)*(r^2 - 9)) - 720/(r^2 - 4) - 6480/((r^2 - 4)*(r^2 - 9)) 5 0] 2: [ 0 0 1 0 0] [ 0 2 (r + 4)*(r - 4) 10 0] [ 1 (r + 4)*(r - 4) 1/2*r^4 - 25/2*r^2 + 108 5*(r + 4)*(r - 4) 5] [ 0 10 5*(r + 4)*(r - 4) 50 0] [ 0 40*r^2/((r^2 - 4)*(r^2 - 9)) - 40/(r^2 - 4) - 360/((r^2 - 4)*(r^2 - 9)) 5 200*r^2/((r^2 - 4)*(r^2 - 9)) - 200/(r^2 - 4) - 1800/((r^2 - 4)*(r^2 - 9)) 0] 3: [ 0 0 0 1 0] [ 0 0 (r + 3)*(r - 3) 4 1] [ 0 (r + 3)*(r - 3) 1/2*(r + 4)*(r + 3)*(r - 3)*(r - 4) 5*(r + 3)*(r - 3) 0] [ 1 4 5*(r + 3)*(r - 3) 16 4] [ 0 1 5*r^6/((r^2 - 4)*(r^2 - 9)) - 5*r^4/(r^2 - 4) - 170*r^4/((r^2 - 4)*(r^2 - 9)) + 125*r^2/(r^2 - 4) + 1845*r^2/((r^2 - 4)*(r^2 - 9)) - 720/(r^2 - 4) - 6480/((r^2 - 4)*(r^2 - 9)) 4 0] 4: [ 0 0 0 0 1] [ 0 0 0 (r + 2)*(r - 2) 0] [ 0 0 1/2*(r + 3)*(r + 2)*(r - 2)*(r - 3) 0 0] [ 0 (r + 2)*(r - 2) 0 4*(r + 2)*(r - 2) 0] [ 1 4*r^4/((r^2 - 4)*(r^2 - 9)) - 4*r^2/(r^2 - 4) - 52*r^2/((r^2 - 4)*(r^2 - 9)) + 16/(r^2 - 4) + 144/((r^2 - 4)*(r^2 - 9)) 5*r^6/((r^2 - 4)*(r^2 - 9)) - 5*r^4/(r^2 - 4) - 110*r^4/((r^2 - 4)*(r^2 - 9)) + 65*r^2/(r^2 - 4) + 765*r^2/((r^2 - 4)*(r^2 - 9)) - 180/(r^2 - 4) - 1620/((r^2 - 4)*(r^2 - 9)) 16*r^4/((r^2 - 4)*(r^2 - 9)) - 16*r^2/(r^2 - 4) - 208*r^2/((r^2 - 4)*(r^2 - 9)) + 64/(r^2 - 4) + 576/((r^2 - 4)*(r^2 - 9)) 4]
The intersection numbers can be checked to be nonnegative and integral for odd values of $r \ge 5$.
p.pTable(factor = True, simplify = True)
0: [ 1 0 0 0 0] [ 0 5/24*(r + 1)*(r - 1)*(r - 2)*r 0 0 0] [ 0 0 1/24*(r + 3)*(r + 2)*(r + 1)*(r - 2) 0 0] [ 0 0 0 5/24*(r + 2)*(r + 1)*(r - 1)*r 0] [ 0 0 0 0 1/24*(r + 2)*(r - 1)*(r - 2)*(r - 3)] 1: [ 0 1 0 0 0] [ 1 1/12*(r^2 - 3*r + 6)*(r + 1)*(r - 1) 1/48*(r + 3)*(r + 2)*(r + 1)*(r - 3) 1/12*(r + 2)*(r + 1)*(r - 1)*(r - 3) 1/48*(r + 2)*(r - 1)*(r - 3)*(r - 5)] [ 0 1/48*(r + 3)*(r + 2)*(r + 1)*(r - 3) 0 1/48*(r + 3)*(r + 2)*(r + 1)*(r - 1) 0] [ 0 1/12*(r + 2)*(r + 1)*(r - 1)*(r - 3) 1/48*(r + 3)*(r + 2)*(r + 1)*(r - 1) 1/12*(r + 3)*(r + 2)*(r + 1)*(r - 1) 1/48*(r + 2)*(r + 1)*(r - 1)*(r - 3)] [ 0 1/48*(r + 2)*(r - 1)*(r - 3)*(r - 5) 0 1/48*(r + 2)*(r + 1)*(r - 1)*(r - 3) 0] 2: [ 0 0 1 0 0] [ 0 5/48*(r + 1)*(r - 1)*(r - 3)*r 0 5/48*(r + 1)*(r - 1)^2*r 0] [ 1 0 1/48*(r^2 + 3*r - 12)*(r + 5)*(r + 1) 0 1/48*(r + 4)*(r - 1)^2*(r - 3)] [ 0 5/48*(r + 1)*(r - 1)^2*r 0 5/48*(r + 5)*(r + 1)*(r - 1)*r 0] [ 0 0 1/48*(r + 4)*(r - 1)^2*(r - 3) 0 1/48*(r + 1)*(r - 1)*(r - 3)*(r - 4)] 3: [ 0 0 0 1 0] [ 0 1/12*(r + 1)*(r - 1)*(r - 2)*(r - 3) 1/48*(r + 3)*(r + 1)*(r - 1)*(r - 2) 1/12*(r + 3)*(r + 1)*(r - 1)*(r - 2) 1/48*(r + 1)*(r - 1)*(r - 2)*(r - 3)] [ 0 1/48*(r + 3)*(r + 1)*(r - 1)*(r - 2) 0 1/48*(r + 5)*(r + 3)*(r + 1)*(r - 2) 0] [ 1 1/12*(r + 3)*(r + 1)*(r - 1)*(r - 2) 1/48*(r + 5)*(r + 3)*(r + 1)*(r - 2) 1/12*(r^2 + 3*r + 6)*(r + 1)*(r - 1) 1/48*(r + 3)*(r - 1)*(r - 2)*(r - 3)] [ 0 1/48*(r + 1)*(r - 1)*(r - 2)*(r - 3) 0 1/48*(r + 3)*(r - 1)*(r - 2)*(r - 3) 0] 4: [ 0 0 0 0 1] [ 0 5/48*(r + 1)*(r - 1)*(r - 5)*r 0 5/48*(r + 1)^2*(r - 1)*r 0] [ 0 0 1/48*(r + 4)*(r + 3)*(r + 1)*(r - 1) 0 1/48*(r + 3)*(r + 1)^2*(r - 4)] [ 0 5/48*(r + 1)^2*(r - 1)*r 0 5/48*(r + 3)*(r + 1)*(r - 1)*r 0] [ 1 0 1/48*(r + 3)*(r + 1)^2*(r - 4) 0 1/48*(r^2 - 3*r - 12)*(r - 1)*(r - 5)]
We now compute the triple intersection numbers with respect to three vertices $x, y, z$ mutually in relation $1$. Note that we have $p^1_{11} = (r^2 - 3r + 6)(r^2 - 1)/12 > 0$ for all $r \ge 5$, so such triples must exist. The parameter $A$ will denote the number of vertices in relations $1, 2, 3$ to $x, y, z$, respectively.
S111 = p.tripleEquations(1, 1, 1, params={'A': (1, 2, 3)})
S111[1, 1, 1]
1/16*(r^5 + 2*r^3 - 3*(16*A + 1)*r - 18*r^2 + 18)/r
The above triple intersection number is integral for odd values of $r$ whenever $5r + 4 - 9/r$ is divisible by $8$.
(S111[1, 1, 1] + (5*r + 4 - 9/r)/8).subs(r == 2*t + 1).factor()
t^4 + 2*t^3 + 2*t^2 - 3*A
The above expression is integral only when $r$ divides $9$. As we must have $r \ge 5$, we conclude that a $Q$-polynomial association scheme with Krein array $\{r^2 - 4, r^2 - 9, 10, 1; 1, 2, r^2 - 9, r^2 - 4\}$ and $r \ne 9$ does not exist. Consequently, no tight $4$-design in $H((9a^2+1)/5, 6)$ exists, thus completing the classification of orthogonal arrays of strength $4$ meeting the Rao bound.
The case $r = 9$ with Krein array $\{77, 72, 10, 1; 1, 2, 72, 77\}$ remains feasible. Such a scheme would have a strongly regular graph with parameters $(v, k, \lambda, \mu) = (540, 154, 28, 50)$ as a subscheme. This parameter set is feasible, however no example is known.