Existence of unresolved small-order abelian near-factorizations

Determine whether λ-fold near-factorizations exist for the abelian groups and parameter quadruples listed in Table 3, whose nonexistence is established only by exhaustive computer search rather than by theoretical results.

Background

The paper develops necessary conditions and constructions for λ-fold near-factorizations, then uses exhaustive computation for abelian groups of order at most 35. Table 3 records parameter sets for which the computational search found no near-factorization.

These cases remain unresolved from a theoretical perspective: the computations establish nonexistence for the listed finite instances, but the paper does not provide theoretical nonexistence proofs explaining those results. The table includes, among others, cases in cyclic groups and noncyclic abelian groups with specified values of |G|, |S|, |T|, and λ.