A family of k -dimensional subspaces of Fqn, which forms a partial spread, is called L -almost affinely disjoint (or briefly [n, k, L]q-AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family. The notion of AADs was introduced and its connections with some problems in coding theory were presented by Liu et al. (2021) in . In particular, Liu et al. investigated the size of maximal such sets and presented a conjecture. They also proved that their conjecture is true for k = 1 and k = 2 when L is sufficiently large. In this work, we give some constructions of large AADs for n > 3k, hence & BULL; improve the lower bound of Liu et al. and & BULL; prove that their conjecture is still true for k > 3 when n = 3k. Our constructions are based on the usage of some certain maximum rank distance codes together with a modified version of the lifting idea and linkage construction.(c) 2022 Elsevier Inc. All rights reserved.