The theory of $(\lambda ,\mu )$-fold factorizations of groups

Abstract

Motivated by recent work on $\lambda$-fold near-factorizations by Kreher, Li, and Stinson (Designs, Codes and Cryptography, 2026), we introduce and systematically investigate $(\lambda ,\mu )$-fold factorizations as combinatorial structures with prescribed representation multiplicities. Such a factorization consists of two subsets of a finite group whose products represent each nonidentity element exactly $\lambda$ times and the identity exactly $\mu$ times. These structures generalize $\lambda$-fold near-factorizations and provide a common framework for several families arising in combinatorial design theory, including strong external difference families with two equal-sized subsets, as well as binary sequence pairs with two-level correlation. Their associated $\{0,1\}$-matrices satisfy the Bridges-Ryser equation $XY=\lambda (J-I)+\mu I$, linking the factorization problem to a classical matrix framework for combinatorial designs. We establish the foundational theory of $(\lambda ,\mu )$-fold factorizations, provide two general constructions, and prove several nonexistence results for finite abelian groups.

arXiv commentUpdated abstract, minor revisions, and several additional references