Sudoku Analogues of Baranyai's Theorem

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 $d$