Exponential growth of inequivalent codes from bent vectorial functions

Let $m$ be a positive integer and let $1\leq \ell\leq m$. A $(2m,\ell)$ bent vectorial function is used to construct a binary code with parameters

$$[2^{2m},\,2m+1+\ell,\,2^{2m-1}-2^{m-1}].$$

Two codes are inequivalent when they are not equivalent as binary codes, and a code admits a 2-transitive automorphism group when its automorphism group acts 2-transitively on its coordinate positions.

Exponential-growth conjecture. For any given $\ell$ in the range $1\leq \ell\leq m$, the number of inequivalent codes with these parameters obtained from $(2m,\ell)$ bent vectorial functions via the stated construction grows exponentially with linear growth of $m$, and most of these codes do not admit a 2-transitive automorphism group.

The conjecture is motivated by examples giving several inequivalent codes with these parameters and by further evidence in the paper. It predicts both abundant inequivalence and the typical absence of 2-transitive automorphism groups as the number of variables grows.