Existence characterization for affine resolvable semi-regular group divisible designs

Characterize the parameter values for which affine resolvable semi-regular group divisible designs with lambda_1=0 and block size k=m exist, beyond the necessary structural conditions established in the paper.

Background

The main construction derives full pairwise additive BIB designs from affine resolvable semi-regular group divisible designs (ARSRGD designs) with lambda_1=0 and k=m. The paper notes that these designs are equivalent to a class of orthogonal arrays and uses a known characterization of affine resolvability in terms of the conditions b=v-m+r and k^2/n being an integer.

Although those conditions characterize when a resolvable semi-regular group divisible design is affine resolvable, they do not settle existence of the underlying designs for all parameter values. Thus, determining the existence spectrum of the relevant ARSRGD designs remains an unresolved supporting problem for expanding the construction method.