Research Information System
Discrete Mathematics, vol.349, no.9, 2026 (SCI-Expanded, Scopus)
Let Fq denote the finite field of size q and Fqn denote the standard vector space over Fq. A collection of k -dimensional subspaces of Fqn with pairwise trivial intersection is called a partial k-spread . A partial k -spread is called an almost affinely disjoint (AAD) subspace family if each affine coset of a member of this family intersects at most L subspaces in the family. In particular, if n=2k+1, the family is called L-nice . Arıkan et al. [1] gave a construction of large AAD families of size q for L -nice families with the parameters [n=2k+1,k,k]q. In their construction, the authors used the lifting idea and the linkage construction algebraic techniques rather than Reed-Solomon codes and finite geometry. They showed that their construction is asymptotically optimal for k=2, k=3, and conjectured that this construction is still asymptotically optimal for any k>3 where L=k. Later on, in [2] , Düzgün et al. proved that the conjecture made by Arıkan et al. is true for k>3. In this paper, we provide an alternative proof for the same conjecture for all k>3, using a different approach than that given in [2] . As a result of this new approach, we are able to generate [n=t(2k+1),k′=tk,L=k]q-AAD families of size qt with the help of block matrices. To achieve this, we generalize the determinant of a matrix derived from the constructions for k=2 and k=3 given in [1] . Specifically, our construction also proves the conjecture given in [12] for some n values in the range 2k′