Sudoku Analogues of Baranyai's Theorem

Abstract: Motivated by higher-dimensional generalizations of Sudoku, we study exact block-structured decompositions, algebraic characterizations, and orthogonality for Sudoku hypercubes. Let $n=\prod_{i=1}^d a_i$, let $b_i=n/a_i$, and consider the $λ$-fold complete $d$-uniform $d$-partite hypergraph with $d$ vertex classes of size $n$, where the $i$th class is partitioned into $a_i$ groups of size $b_i$. Given positive integers $m_1,\dots,m_k$ with $\sum_{j=1}^k m_j=λn<sup>d$, we partition the edges into color classes of sizes $m_1,\dots,m_k$ so that, in color $j$, vertex degrees and block counts are each either $\lfloor m_j/n\rfloor$ or $\lceil m_j/n\rceil$, while the multiplicity of an underlying edge is either $\lfloor m_j/n<sup>d\rfloor$ or $\lceil m_j/n<sup>d\rceil$. When $m_j=nr_j$, the vertex and block balances are exact, yielding block factorizations and higher-dimensional Sudoku analogues of Baranyai's theorem. Within the same block framework, we give a Delsarte characterization of the Sudoku condition using association schemes and study mutually orthogonal Sudoku hypercubes of order $q<sup>3$ for prime powers $q$. For block sizes $(q<sup>3,q<sup>2,q)$ and $(q<sup>3,q<sup>3,1)$, the resulting families attain a general upper bound and are best possible. For block size $(q<sup>2,q<sup>2,q<sup>2)$, we construct $q<sup>2(q<sup>2-1)(q<sup>2-q)$ mutually orthogonal hypercubes; this construction is asymptotically best possible as $q\to\infty$.