# Affine automorphisms of algebraic function field towers

Are there any well-known towers of function fields over finite fields whose automorphism groups contain a transitive subgroup consisting solely of affine maps?

For a (non)example of what I'm looking for: let $\mathbb{F}$ be the finite field of $q^2$ elements, and consider the Hermitian function field on the Hermitian curve in $\mathbb{F}^2$ defined by $x^{q+1} = y^q + y$. The automorphisms here contain a subgroup of maps $(x,y) \mapsto (ax+b, a^{q+1}y + ab^qx + c)$ for $a \in \mathbb{F}^*$, $b,c \in \mathbb{F}$ such that $b^{q+1} = c^q +c$. These maps are clearly affine (degree at most one) and it can be shown that this subgroup is transitive on the $q^3$ points of the Hermitian curve.

However, now consider the Hermitian tower, where we now work over a curve in $\mathbb{F}^m$ consisting of points $(x_1,\ldots,x_m)$ satisfying $x_i^{q+1} = x_{i+1}^q + x_{i+1}$ for $i=1,\ldots,m-1$. If I want to map $x_1 \mapsto ax_1+b$, then $x_2$ is forced to map to $a^{q+1}x_2 + ab^qx_1 + c$ for some $c^q + c = b^{q+1}$, which is fine, but then this forces a map on $x_3$ which is not affine.

Edit: I'm interested in examples of towers with lots of rational points over the finite field in question. The application I have in mind is constructing error correcting codes with large block-length, which is the number of rational points.

By "transitive" you seem to mean transitive on the set of $\mathbb{F}$-rational points, for some finite field $\mathbb{F}$. There are lots of towers of function fields having the property you requested: for instance, just pick any tower which doesn't have any rational points over the finite field under consideration.
(Incidentally, the Hermitian curve has $q^3+1$ points over the field of cardinality $q^2$, and the group of automorphisms which you listed actually has two orbits on the set of these points.)