Existence of a symmetric transversal design $\mathrm{STD}_2[12;6]$

A symmetric transversal design $\mathrm{STD}_\lambda[k;u]$ is an incidence structure of $ku$ points partitioned into $k$ point classes of $u$ points, with $ku$ blocks of size $k$ each meeting every point class exactly once, in which two points from different point classes lie in exactly $\lambda$ common blocks, and whose dual structure satisfies the same axioms (necessarily $k = \lambda u$). Every known example has prime-power point-class size $u$; whether any STD with non-prime-power $u$ exists is a broader background question, and the smallest parameter set for which neither a construction nor a nonexistence proof is known is $\mathrm{STD}_2[12;6]$ (72 points, 12 classes of 6, $\lambda = 2$). The question is linked to the smallest undecided order for projective planes: a projective plane of order 12 admitting a collineation of order 2 (necessarily an involutory elation) contracts to an $\mathrm{STD}_2[12;6]$. Current knowledge tightly constrains both sides: any automorphism group of a putative $\mathrm{STD}_2[12;6]$ is a $\{2,3\}$-group (Suetake 2009), and any collineation group of a projective plane of order 12 has order 1, 2, or 3 (Janko-Van Trung 1982; Akiyama-Suetake-Tanaka 2019, 2023). Whether $\mathrm{STD}_2[12;6]$ exists is open.

Background

A transversal design is an incidence structure of points and blocks whose point set is partitioned into 'point classes' (also called groups) so that every block is incident with exactly one point of each class, and so that any two points from distinct classes are jointly incident with exactly $\lambda$ blocks, where $\lambda$ is the index of the design. A symmetric transversal design $\mathrm{STD}_\lambda[k;u]$ is a transversal design that is self-dual in a strong sense: counting forces $k = \lambda u$, the number of blocks equals the number of points ($ku$), and the dual structure, obtained by interchanging the roles of points and blocks, satisfies the same axioms; equivalently, the block set is partitioned into $k$ block classes of $u$ blocks so that each point is incident with exactly one block of each block class and two blocks from different block classes meet in exactly $\lambda$ points. (This is Definition 2.7 of Akiyama-Suetake-Tanaka, Australasian Journal of Combinatorics 74 (2019), in the notation used throughout Suetake's series of papers.) For $\mathrm{STD}_2[12;6]$ the numbers are: 72 points in 12 point classes of 6 points; 72 blocks, each incident with 12 points, one from every point class; every two points from different point classes jointly incident with exactly 2 blocks; and the dual structure again a transversal design with 12 block classes of 6 blocks.

The theory is inseparable from finite projective planes. A projective plane of order $n$ is an incidence structure of $n^2 + n + 1$ points and equally many lines, each line incident with $n + 1$ points, any two points on exactly one line and any two lines meeting in exactly one point. Planes coordinatized by finite fields exist for every prime-power order $n$; the Bruck-Ryser theorem, a number-theoretic obstruction which forces $n$ to be a sum of two squares whenever $n \equiv 1$ or $2 \pmod 4$, excludes many other orders, but for $n = 12$ (which is $0 \bmod 4$) it gives no information. Order 10 was famously eliminated by the computer-assisted computation of Lam-Thiel-Swiercz (1989), which determined the weight enumerator (the numbers of codewords of each weight) of the code generated by the rows of any putative plane's incidence matrix, so order 12 is the smallest order whose existence is undecided. Akiyama-Suetake-Tanaka (2019) record the classical bridge: every projective plane of order $n$ contains an $\mathrm{STD}_1[n;n]$ as a substructure (fix a flag, that is, an incident point $r_0$ and line $L_0$; the points off $L_0$ form the point set and the lines not through $r_0$ the blocks, with point classes given by the $n$ lines through $r_0$ other than $L_0$ and block classes by the $n$ points of $L_0$ other than $r_0$), and conversely every $\mathrm{STD}_1[n;n]$ extends uniquely, up to isomorphism, to a projective plane of order $n$. Thus the case $\lambda = 1$, $u = 12$ of the general STD question is exactly the order-12 plane problem.

For larger $\lambda$ the designs detach from planes, but all known constructions still produce prime-power point-class size $u$. An STD is called class regular when it admits an automorphism group acting regularly (sharply transitively) on each point class; this relates $\mathrm{STD}_\lambda[\lambda u;u]$ to generalized Hadamard matrices over a group of order $u$ — square matrices of order $\lambda u$ whose entries lie in the group and whose entrywise quotients of any two distinct rows list every group element exactly $\lambda$ times — and the known constructions of such matrices run over groups of prime-power order; the spread-type constructions of Hiramine (2011), built from spreads (partitions of the point set of a finite projective space into disjoint subspaces of equal size), likewise have class size $q$, a prime power. For $u = 3$ the spectrum is largely mapped: writing $n_\lambda$ for the number of nonisomorphic $\mathrm{STD}_\lambda[3\lambda;3]$, one knows $n_1 = 1$, $n_2 = 1$, $n_3 = 4$, $n_4 = 1$, $n_5 = 0$, $n_6 \ge 20$, $n_7 \ge 5$, $n_8 \ge 24$ (Suetake 2005; Akiyama-Ogawa-Suetake 2009; Hatono-Suetake 2010). Since all known constructions have prime-power class size, a broader background question arises: does any symmetric transversal design $\mathrm{STD}_\lambda[k;u]$ with $u$ not a prime power exist? The smallest parameter set not settled by known results is $\mathrm{STD}_2[12;6]$: $u = 6$ is the smallest non-prime-power, the case $\lambda = 1$ is impossible classically (a transversal design with 6 groups of size 6 and index 1 would amount to 4 mutually orthogonal Latin squares of order 6, and no two orthogonal Latin squares of order 6 exist, by Tarry's resolution of Euler's thirty-six officers problem), while $\lambda = 2$ is open.

$\mathrm{STD}_2[12;6]$ is also the design-theoretic face of the involution problem for planes of order 12. A collineation is an incidence-preserving permutation of the points and lines of a plane. An involutory collineation of a plane of order 12 must be an elation, that is, it fixes some line (the axis) pointwise together with a point on that line (the center): an involutory homology (center off the axis) is possible only in odd order, since on any line through its center the $n - 1$ moved points must be swapped in pairs, and a Baer involution, which fixes pointwise a subplane of order $\sqrt{n}$, requires $n$ to be a square, which 12 is not. Taking the orbits of such an involutory elation on the points off the axis (the 12 affine lines through the center give the 12 point classes, of 6 two-point orbits each) and the orbits of the lines not through the center (giving 72 blocks of 12 point-orbits) contracts the plane to an $\mathrm{STD}_2[12;6]$. This contraction mechanism, developed by Hiramine-Suetake (2008), powers the nonexistence program for symmetric planes: Akiyama-Suetake (2008) proved that no $\mathrm{STD}_2[12;6]$ admits a semiregular automorphism group of order 4 (one fixing no point and no block), and deduced that no plane of order 12 has a collineation group of order 8. On the plane side, Janko-Van Trung (1982) proved that the full collineation group of any plane of order 12 is a $\{2,3\}$-group (a finite group whose order has no prime divisors other than 2 and 3); successive eliminations of possible group orders 6 (1981, 1986), 16 (2004), 8 (2008), 9 (2019), and 4 (2023) now leave only orders 1, 2, and 3. On the design side, Suetake (2009) proved that any automorphism group of a putative $\mathrm{STD}_2[12;6]$ is likewise a $\{2,3\}$-group, using a theorem of Hiramine-Suetake (2008) that an automorphism fixing every point class setwise (an 'elation' of the design) is trivial, then eliminating element orders 5, 7, and 11 by orbit-matrix arguments (orbit matrices are tables recording how blocks meet the orbits of a group action) and intersection counting. Consequently, neither the putative design nor a plane producing it can carry appreciable symmetry, and both the construction and the refutation of the unconstrained case are wide open.

Problem Statement

Does there exist a symmetric transversal design $\mathrm{STD}_2[12;6]$: an incidence structure of 72 points partitioned into 12 point classes of 6 points, with 72 blocks each incident with exactly 12 points, one from each point class, in which any two points from different point classes are jointly incident with exactly 2 blocks, and whose dual structure satisfies the same axioms (equivalently, the 72 blocks are partitioned into 12 block classes of 6 blocks such that each point is incident with exactly one block of each block class and two blocks from different block classes are jointly incident with exactly 2 points)?

All known examples of symmetric transversal designs have prime-power point-class size $u$, and for $u = 6$ the case $\lambda = 1$ is excluded classically, so $\mathrm{STD}_2[12;6]$ is the first parameter set for which neither a construction nor a nonexistence proof is known. The broader question whether any $\mathrm{STD}_\lambda[k;u]$ with non-prime-power $u$ exists is background context, not the target here: a construction at other parameters would not resolve this problem. The problem is linked to, but strictly weaker than, the involution question for projective planes: a projective plane of order 12 admitting a collineation of order 2 — necessarily an involutory elation — contracts to an $\mathrm{STD}_2[12;6]$, while no converse implication is known.

The answer is affirmative, given by an explicitly presented incidence structure satisfying the stated axioms, or negative, given by a proof that no such design exists; a nonexistence proof would also rule out a collineation of order 2 in a projective plane of order 12. No symmetry is assumed of the design: the question concerns unconstrained incidence structures satisfying the stated axioms.

References

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