Almost affinely disjoint subspaces and covering Grassmannian codes.

A family of k -dimensional subspaces of F q n with pairwise trivial intersection is called L -almost affinely disjoint (AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family. Liu et al. introduced the notion of AAD family, and investigated the lower and upper bounds of maximal such sets and conjectured that for any k , n and a large enough L = L (n , k) , there exists an [ n , k , L ] q -AAD family with size q n − 2 k in [Finite Fields Appl. 75 (2021), 101879]. Etzion and Zhang introduced covering Grassmannian codes (CGCs) for generalized combination networks [IEEE Trans. Inf. Theory, 65 (2019), 4131–4142.]. An α - (n , k , δ) q c covering Grassmannian code C is a set of k -dimensional subspaces of F q n , such that every set of α codewords of C spans a subspace of dimension at least k + δ. In this paper, we give a construction of AAD families and CGCs based on maximum rank metric codes and caps in projective geometries, and a recursive construction based on maximum rank metric codes. As a consequence, we prove that Liu et al.'s conjecture is still true for k ≥ 3 and n = 4 k , and improve lower bounds on maximum sizes of AAD families and lower bounds on maximum sizes of 3- (n , k , 2 k) q c covering Grassmannian codes. [ABSTRACT FROM AUTHOR]