The ternary code support 2-design conjecture

Let $m\geq 3$ be odd, let $v=(3^m-1)/2$, and let $\mathcal{C}_m$ be either of the ternary cyclic codes described in the paper. Let $A_k$ denote the number of codewords of Hamming weight $k$, and define

$$\mathcal{P}=\{0,1,2,\cdots,v-1\}.$$

For $A_k\neq 0$, let $\mathcal{B}$ be the set of supports of the weight-$k$ codewords of $\mathcal{C}_m$. The ternary code support 2-design conjecture. For every such $k$, $(\mathcal{P},\mathcal{B})$ is a $2$-$(v,k,\lambda)$ design. This predicts that the supports of every nonzero weight in these code families form 2-designs; the paper gives the code constructions and relates the weight-$4$ case to a Steiner-system construction, but the conjecture is not resolved here.