Base-case spectrum of the Wang–Shi lifting for large sets of Kirkman triple systems: existence of LKTS(q+2) for prime powers q ≡ 1 (mod 6), and canonicity of the lifted large set

The Wang–Shi lifting constructs a large set of Kirkman triple systems LKTS($q^n+2$) from any base LKTS($q+2$) whenever $q$ is a prime power congruent to $1$ modulo $6$, thereby reducing the existence problem for this infinite family of orders to the base case $n=1$. The reduction leaves two questions open. First, the complete characterization of which prime powers $q \equiv 1 \pmod 6$ admit a base LKTS($q+2$): after the direct constructions of Zheng, Chang and Zhou, the only unresolved prime powers below $400$ are $q = 373$ and $q = 397$ (orders $375$ and $399$), and essentially every prime power $q \geq 401$ is likewise unresolved; no nonexistence result is known for any admissible order. Second, whether, when the base large set is not translation-invariant, the isomorphism type of the lifted large set depends on the auxiliary choices of coset representatives and skew-symmetric bilinear forms made inside the construction — a question on which no published work exists beyond the source paper's translation-invariant special case.

Background

A Steiner triple system of order $v$, STS($v$), is a pair $(X,\mathcal{B})$ where $X$ is a $v$-element point set and $\mathcal{B}$ is a collection of $3$-element subsets, called blocks or triples, such that every $2$-element subset of $X$ lies in exactly one block. A parallel class is a set of blocks that partitions $X$; an STS($v$) whose block set can be partitioned into parallel classes is called resolvable, and a resolvable STS($v$) is known as a Kirkman triple system, KTS($v$). A KTS($v$) exists if and only if $v \equiv 3 \pmod 6$. Two STS($v$) on the same point set are disjoint if they share no block, and by a counting argument at most $v-2$ of them can be pairwise disjoint; a maximum family of $v-2$ pairwise disjoint STS($v$) necessarily partitions all $\binom{v}{3}$ triples of the point set and is called a large set of Steiner triple systems, LSTS($v$). If moreover every constituent is resolvable, one obtains a large set of Kirkman triple systems, LKTS($v$), the object arising from Sylvester's extension of Kirkman's 1850 schoolgirl problem. The existence problem for LSTS($v$) was completely settled in the 1980s–90s by Lu and Teirlinck, but the corresponding problem for LKTS($v$) remains far from settled: what is known is a sparse collection of direct constructions for specific orders, surveyed by Chang and Zhou, together with product-type and other recursive constructions.

Wang and Shi (2017) contributed a recursion of a new kind. Let $q \equiv 1 \pmod 6$ be a prime power and suppose a base LKTS($q+2$) exists on the point set $\mathbb{F}_q \cup \{\infty_1,\infty_2\}$. Their Theorem 2.1 constructs an LKTS($q^n+2$) on $W \cup \{\infty_1,\infty_2\}$, where $W = \mathbb{F}_q^n$ is viewed as an $n$-dimensional vector space over $\mathbb{F}_q$, for every integer $n \geq 2$. The construction is organized around a large set of 'frames' over $W$ that is invariant under translations of $W$; the frame triples are $\{x,y,-x-y\}$ for linearly independent $x,y \in W$, and the holes (the $1$-dimensional affine subspaces of $W$) are filled using, for each of certain subgroups $K_i$, a system $R_i$ of representatives of the cosets of $K_i$ together with an associated function $p_i : W \to \mathbb{F}_q$, and a family of skew-symmetric bilinear forms $f_L : L \times L \to \mathbb{F}_q$, one for each $2$-dimensional subspace $L$ of $W$, which governs how the zero-sum triples of $W$ are partitioned. The base LKTS($q+2$) is the only design-theoretic input, so the existence problem for the entire family of orders $\{q^n+2 : n \geq 2\}$ reduces to the base case $n=1$.

The paper records that an LKTS($q+2$) 'is known for $q = 7, 13, 19, 25, 31, 37, 43, 49, 61$ and many more values', but it proves no general base-case theorem, and later work (refs 2, 3) has filled in most — but not all — of the range below $400$ while leaving the general question untouched. The paper also proves only a partial canonicity statement, in its Remark 3.7: if the base large set is translation-invariant, meaning its constituent systems satisfy $\mathcal{D}_i = \mathcal{D}_0 + i$ and their parallel classes $Q_{i,j}$ satisfy $Q_{i,j} = Q_{0,j} + i$ for all $i$ and $j$, then the output large set is again translation-invariant and a different choice of the representatives $R_i$ 'gives the same design in this case'. Outside this translation-invariant situation, nothing is said about whether the isomorphism type of the output depends on the choices of the $R_i$ or of the forms $f_L$.

Problem Statement

Determine the base-case existence spectrum of the Wang–Shi lifting, and the canonicity of its output. For prime powers $q \equiv 1 \pmod 6$:

(1) Primary objective — characterize completely the values of $q$ for which a base LKTS($q+2$) exists. In its current shape this asks, first, for the two smallest unresolved cases: an LKTS($375$) (for $q = 373$) or a proof that none exists, and likewise an LKTS($399$) (for $q = 397$); and, beyond them, existence or nonexistence for arbitrary prime powers $q \equiv 1 \pmod 6$ with $q \geq 401$.

(2) Secondary objective, on the same construction — when the base LKTS($q+2$) is not translation-invariant, decide whether the isomorphism type of the LKTS($q^n+2$) produced by the Wang–Shi construction depends on the auxiliary choices made inside it, namely the systems $R_i$ of coset representatives of the subgroups $K_i$ (which determine the associated functions $p_i$) and the family of skew-symmetric bilinear forms $f_L$ on the $2$-dimensional subspaces of $W = \mathbb{F}_q^n$: either prove that the output is determined up to isomorphism for every non-translation-invariant base and every $n \geq 2$ regardless of these choices, or exhibit one base LKTS($q+2$), one $n \geq 2$, and two choices of $(R_i, f_L)$ whose two outputs are non-isomorphic large sets of order $q^n+2$ (two large sets being isomorphic if some bijection between their point sets carries every constituent Kirkman triple system of one onto a constituent Kirkman triple system of the other).

The two strands concern the same lifting package: strand (1) supplies its admissible inputs and strand (2) asks whether the output is canonical. The case already settled by the source's Remark 3.7 — choice-independence of the representatives $R_i$ when the base is translation-invariant — is excluded from (2) as known.

References

1. Chen Wang, Cong Shi, Large sets of Kirkman triple systems with order q^n + 2, Discrete Mathematics 340(5):1042–1045, 2017. DOI 10.1016/j.disc.2016.11.008, https://doi.org/10.1016/j.disc.2016.11.008 (also arXiv:1307.3019, https://arxiv.org/abs/1307.3019)
2. Hao Zheng, Yanxun Chang, Junling Zhou, Direct constructions of large sets of Kirkman triple systems, Designs, Codes and Cryptography 83(1):23–32, 2017 (online 2016). DOI 10.1007/s10623-016-0197-4, https://doi.org/10.1007/s10623-016-0197-4
3. Hao Zheng, Yanxun Chang, Junling Zhou, Large sets of Kirkman triple systems of prime power sizes, Designs, Codes and Cryptography 85(3):411–423, 2017 (online 2016). DOI 10.1007/s10623-016-0315-3, https://doi.org/10.1007/s10623-016-0315-3
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6. Yan Liu, Jianguo Lei, More results on large sets of Kirkman triple systems, Designs, Codes and Cryptography 91(8):2677–2686, 2023. DOI 10.1007/s10623-023-01221-9, https://doi.org/10.1007/s10623-023-01221-9
7. Yuli Tan, Junling Zhou, Steiner quadruple systems with minimum colorable derived designs: constructions and applications, Designs, Codes and Cryptography 93(10):4117–4140, 2025. DOI 10.1007/s10623-025-01677-x, https://doi.org/10.1007/s10623-025-01677-x
8. Chen Wang, A new recursive construction for large sets of Kirkman triple systems, preprint, 2026. arXiv:2607.21521, https://arxiv.org/abs/2607.21521
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