Asymptotically optimal [2k+1,k,k]q-almost affinely disjoint subspaces

Official URL: https://dx.doi.org/10.1016/j.disc.2024.114140

Let Fq denote the finite field of size q and Fqn denote the set of n-tuples of elements from Fq. 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. Polyanskii and Vorobyev introduced AAD subspace families in [23] (2019) in order to construct some types of primitive batch codes. For this purpose, they made use of Reed-Solomon codes and hence they presented [n=2k+1,k,L=k]-AAD subspace families of size ⌊q/k⌋. Later on, AAD spaces received a notable attention and have been studied for various parameters, see Liu et al. [18] (2021) and Otal and Arıkan [21] (2022) for example. In Arıkan et al. (2023) [1], we presented a construction of [2k+1,k,k]-AAD subspace families of size q, which improves the lower bound ⌊q/k⌋ given by Polyanskii and Vorobyev in [23] (2019) k times. We also proved that our construction yields AAD subspace families for k=2 and k=3, and left the proof of the case k≥4 as a conjecture. In this paper, we now prove this conjecture by a newer and more abstract approach. For the proof, we carefully utilize several sophisticated arguments coming from coding theory, algebraic geometry, and linear algebra. In particular, we interpret each subspace as a suitable generator matrix (the lifting idea in random network coding), then algebraically manipulate their AAD-ness relation and produce suitable Van der Monde matrices (the fundamental tools for the BCH bound in coding theory), later apply Cramer's rule (a well-known key tool in linear algebra), and lastly obtain low-degree polynomials as contradictions (the trick coming from algebraic geometry utilized for Goppa codes). We present our constructive proof step-by-step to make it more easy to understand.