An asymptotically optimal construction of almost affinely disjoint subspaces

Abstract

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 [27]. 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•improve the lower bound of Liu et al. and•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.