The theory of (λ,μ)-fold factorizations of groups
Abstract
Motivated by recent work on λ-fold near-factorizations by Kreher, Li, and Stinson (Designs, Codes and Cryptography, 2026), we introduce and systematically investigate (λ,μ)-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 λ times and the identity exactly μ times. These structures generalize λ-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=λ(J−I)+μI, linking the factorization problem to a classical matrix framework for combinatorial designs. We establish the foundational theory of (λ,μ)-fold factorizations, provide two general constructions, and prove several nonexistence results for finite abelian groups.
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BibTeX
@misc{kreher2026theory,
title = {{The theory of $(λ,μ)$-fold factorizations of groups}},
author = {Donald L. Kreher and Shuxing Li and Douglas R. Stinson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15826},
eprint = {2610.15826},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15826},
}
RIS
TY - UNPB
TI - The theory of (λ,μ)-fold factorizations of groups
AU - Donald L. Kreher
AU - Shuxing Li
AU - Douglas R. Stinson
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.15826
UR - https://arxiv.org/abs/2610.15826
AB - Motivated by recent work on λ-fold near-factorizations by Kreher, Li, and Stinson (Designs, Codes and Cryptography, 2026), we introduce and systematically investigate (λ,μ)-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 λ times and the identity exactly μ times. These structures generalize λ-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=λ(J−I)+μI, linking the factorization problem to a classical matrix framework for combinatorial designs. We establish the foundational theory of (λ,μ)-fold factorizations, provide two general constructions, and prove several nonexistence results for finite abelian groups.
ER -