Carpenter–Tonchev rank conjecture for designs from maximal arcs

Let $t\geq 2$, and let $\mathcal{D}$ be a $2-(2^{2t-1}-2^{t-1},2^{t-1},1)$ design with incidence matrix $A$. Carpenter–Tonchev's rank conjecture. The binary rank of $A$ satisfies

$$\operatorname{rank}_{2}(A)\geq 3^t-2^t,$$

and equality

$$\operatorname{rank}_{2}(A)=3^t-2^t$$

holds if and only if $\mathcal{D}$ is embeddable as a maximal $(2^{2t-1}-2^{t-1},2^{t-1})$-arc in $PG(2^t,2)$. This strengthens a conjecture attributed to Laurel Carpenter that the binary rank of every design associated with a hyperoval in $PG(2,2^t)$ equals $3^t-2^t$, and generalizes a conjecture of A. E. Brouwer. The computations in the paper support the conjecture for the 93 designs arising from hyperovals in $PG(2,16)$, but the general assertion remains open.