Let ${\mathbb{F}_q}$ denote the finite field of size q and $\mathbb{F}_q^n$ denote the set of n-tuples of elements from ${\mathbb{F}_q}$. A family of k-dimensional subspaces of $\mathbb{F}_q^n$, 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 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 $\left\lfloor {q/k} \right\rfloor $.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 Arıkan [Finite Fields Their Appl. 84 (2022) 102099] gave some constructions of large AAD subspace families for $n \geq 3k$, hence improved the lower bound of Liu et al. and presented asymptotically optimal AAD subspace families for $n = $ $3k$ when $L \geq 1$.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 Arıkan. We highlight that our construction improves the lower bound given by Polyanskii and Vorobyev to q from $\left\lfloor {q/k} \right\rfloor $. Also we express that our method makes use of linear algebraic techniques rather than Reed-Solomon codes and finite geometry.

We give an infinite family of maximum rank distance (MRD) codes, which covers properly the largest known linear MRD code family. Our family contains infinite families of non-linear MRD codes, which are the first non-linear examples for most of the parameters. We also give explicit examples and a table that demonstrates the proportion of linear and non-linear families for some small parameters.