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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.

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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.)

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Are there any examples where the tower has (preferably) lots of rational points over the finite field? –  Alan Guo Mar 6 at 5:05
It depends what you mean by "lots of rational points". For instance, do you require that the tower should be "asymptotically good", in the sense that the number of rational points should grow like a positive constant times the genus as we move out along the steps of the tower? –  Michael Zieve Mar 6 at 15:57
Yes. The application I have in mind is error correcting codes. –  Alan Guo Mar 7 at 0:21
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