  In this paper we study a certain generalization of combinatorial designs
related to almost difference sets, namely the $t$-adesign, which was coined by
Cunsheng Ding in 2015. It is clear that $2$-adesigns are a kind of partially
balanced incomplete block design which naturally arise in many combinatorial
and statistical problems. We discuss some of their basic properties and give
several constructions of $2$-adesigns (some of which correspond to new almost
difference sets, and others of which correspond to new almost difference
families), as well as two constructions of $3$-adesigns. We also discuss some
basic properties of their incidence matrices and codes.
