A new construction of asymptotically optimal almost affinely disjoint spaces

Official URL: https://dx.doi.org/10.1109/ISIT54713.2023.10206484

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 almost affinely disjoint (AAD) subspace families for n = 2k + 1 in [IEEE ISIT 2019 pp. 360-364] using a different language (by saying "L-nice"instead of "[n,k,L]q-AAD") 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 ?. The general notion of almost affinely disjoint (AAD) subspace families was later introduced and connections with some problems in coding theory were presented by Liu et al. in [Finite Fields Their Appl. 75 (2021) 101879]. In particular, the authors gave upper and lower bounds for the size of AAD subspace families and provided asymptotically optimal constructions of such families for k = 1 and k = 2 when L is sufficiently large, where the polynomial growth in q is n - 2k.Later on, Otal and Arikan [Finite Fields Their Appl. 84 (2022) 102099] gave some constructions of large AAD subspace families for n = 3k, hence improved the lower bound of Liu et al. and presented asymptotically optimal AAD subspace families for n = 3k when L = 1.In this paper, we give a construction of large AAD subspace families of size q for n = 2k + 1. We also prove that our construction is asymptotically optimal for k = 2 and k = 3, and conjecture that our construction is still asymptotically optimal for the remaining cases k > 3, where L = k. Our construction is basically a generalization of a special case of the construction of Otal and Arikan. We highlight that our construction improves the lower bound given by Polyanskii and Vorobyev to q from ? q/k ?. Also we express that our method makes use of linear algebraic techniques rather than Reed-Solomon codes and finite geometry.