2026/07/20 by S. Karthik, Peter J. Cameron, Krishnan Paramasivam
Mathematics · #math.CO
Let X be a finite set of integers with cardinality ν= κλ. A modular balanced partition design is a triplet (X, A, B) satisfying the following conditions: \beginitemize \item A is a partition of X into κ blocks of size λ, such that every element of X appears in exactly one block. If A = \A1, A2, …, Aκ\, then ∑a∈ Ai a ≡ i λ\pmodν, for each i=1,2,…,κ \item B is a partition of X into λ blocks of size κ, such that every element of X appears in exactly one block. If B = \B1, B2, …, Bλ\, then ∑b∈ Bj b ≡ j κ\pmodν, for each j=1,2,…,λ \item Ai ∩ Bj has exactly one element, for any Ai ∈ A, and Bj ∈ B. \itemize We prove the necessary conditions for the existence of a modular balanced partition design. Moreover, we investigate and identify a relationship between a modular balanced partition design and a subgroup magic rectangle. Then by using affine automorphisms of an Abelian group, we prove the existence of non-isomorphic modular balanced partition designs. Finally, we provide a method to construct a transversal design via a modular balanced partition design.