Latin cubes with disjoint subcubes of two orders

TL;DR

The paper addresses the existence problem for latin cubes of order $n$ with pairwise disjoint subcubes of two orders, generalising the classical latin-square realisation problem. It develops two complementary construction methods: an inflation construction that scales existing $\,3 ext{-RP}$ realizations via paired subcubes and a transversal, and a construction from orthogonal arrays to assemble large partial realizations and complete them. The main results establish broad existence for partitions of the form $(a^u b^{k-u})$, notably showing $\,3 ext{-RP}(a^2b^1)$ exists in many cases (with precise bounds on $a$ and $b$) and handling several even/odd congruence classes, while identifying remaining exceptional cases for $a\equiv 3\pmod{6}$. These findings advance the theory of high-dimensional latin cubes, connect to OA-based design methods, and suggest pathways to complete the classification for all partitions.

Abstract

Given a partition $h_1+h_2+\dots+h_k = n$, a latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots ,h_k$ is called a realization. When the values $h_i$ are of at most two sizes, the existence of a realization has been completely determined. However, the existence of a latin cube with pairwise disjoint subcubes of two orders is only partially solved. In this paper, we determine existence for such latin cubes in almost all cases.

Figures (5)

• Figure 1: An order 5 latin cube with disjoint subcubes.
• Figure 2: Two paired realizations: a $\mathop{\mathrm{3-RP}}\nolimits(2^21^1)$ and $\mathop{\mathrm{3-RP}}\nolimits(2^3)$
• Figure 3: The subarray structure of $L$
• Figure 4: Construction of $L$, in two cases depending on $a$. Column and row ordering is shown in margins at the top and left, and dimensions are shown in the top row and left column in bold. All other entries are index values, which together with the dimensions specify the subarray of $K$ used to fill each subarray of $L$.
• Figure 5: A $\mathop{\mathrm{3-RP}}\nolimits(3^3)$

Theorems & Definitions (47)

• Example 1.1
• Theorem 1.2: denes1963some
• Theorem 1.3: heinrich2006latin,heinrich1982disjoint, Theorem 5.1 of kuhl2018latin
• Theorem 1.4: Theorem 3.4 of donovan2025latin
• Theorem 1.5: Theorem 4.4 of donovan2025latin
• Theorem 1.6: Theorem 3.1 of donovan2025latin
• Lemma 1.7: Lemma 4.6 of donovan2025latin
• Corollary 1.8
• Lemma 1.9: Lemma 4.7 of donovan2025latin
• Lemma 1.10: Lemma 4.8 of donovan2025latin