Asymptotically optimal [2<i>k</i>+1, <i>k</i>, <i>k</i>]<i>_q</i>-almost affinely disjoint subspaces

Let F-q denote the finite field of size qand F-q(n) denote the set of n-tuples of elements from F-q. Afamily of k-dimensional subspaces of F-q(n), 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 Lsubspaces 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 left perpendicular q/k right perpendicular. Later on, AAD spaces received a notable attention and have beenstudied for various parameters, see Liu etal. [18](2021) and Otal and Arikan [21](2022) for example. In Arikan et al. (2023) [1], we presented a construction of [2k + 1, k, k]-AAD subspace families of size q, which improves the lower bound left perpendicular q/k right perpendicular given by Polyanskii and Vorobyev in [23](2019) ktimes. We also proved that our construction yields AAD subspace families for k = 2 and k = 3, and left the proof of the case k >= 4as 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 (awell-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. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.