The ternary tenth-power block-size conjecture

Let $m\geq 3$ be odd, and define $k_m$ recursively by

$$k_m=\frac{3^m+1}{2}+3^{m-1}-3k_{m-2},\qquad k_1=2.$$

For $b,c\in {\mathrm{GF}}(3^m)$, let $B_{(x^{10},b,c)}$ denote the corresponding block, and let ${\mathcal{B}}_{(x^{10},k)}$ be the collection of distinct blocks of size $k$. The ternary tenth-power block-size conjecture. The block sizes and multiplicities are

$$|B_{(x^{10},b,c)}|=\begin{cases}\frac{3^m+1}{2}&\text{for }3^m\text{ choices of }(b,c),\k_m&\text{for }(3^m-1)3^m\text{ choices of }(b,c),\end{cases}$$

and

$$|{\mathcal{B}}_{(x^{10},(3^m+1)/2)}|=3^m,\qquad |{\mathcal{B}}_{(x^{10},k_m)}|=\frac{3^m(3^m-1)}{2}.$$

Further, the first incidence structure is a symmetric $2$-$(3^m,(3^m+1)/2,(3^m+1)/4)$ design arising from the squares in ${\mathrm{GF}}(3^m)$, and the second is a $2$-$(3^m,k_m,k_m(k_m-1)/2)$ design. This is described as a fundamental open problem concerning designs supported by monomials over odd-characteristic finite fields.