Two existence results between an affine resolvable SRGD design and a difference scheme (Q1730279)

In this article under review, the authors studied affine resolvable semi-regular group divisible (SRGD) designs and difference schemes. The following main theorem is proved. Let $s$ be a prime or a prime power. Then, the existence of a difference scheme $\mathrm{DS}(sx,s;x)$ implies the existence of an affine resolvable SRGD design with parameters $v=b=xs^{2}$, $r=k=sx$, $\lambda_{1}=0$, $\lambda_{2}=x$, $q=x$; $m=sx$, $n=s$. In [the authors, ibid. 39, No. 2, 293--326 (2009; Zbl 1188.05030)], the same result was proved when $s$ is a prime. In the case when $s$ is a prime, they also found some conditions which make the converse of the theorem holds true.