Asymptotic distance bound conjecture for $[[n,1,d]]$ $M_3$-codes

The distance of an $[[n,1,d]]$ $M_3$-code is the minimum weight of a logical operator, so it is the smallest value $d$ such that $C_d\neq 0$. For code lengths in the indicated congruence classes, let $m$ be defined by $n=6m+1$, $n=6m+3$, or $n=6m+5$.

Distance bound conjecture. The distance of an $[[n,1,d]]$ $M_3$-code satisfies $d<d_{\max}$, with

$$d_{\max}\leq\begin{cases}2m-3 & n=6m+1\geq235,\quad n=6m+3\geq249\\2m-1 & n=6m+5\geq215. \end{cases}$$

This conjecture extends the stronger linear-programming bounds established computationally for smaller values of $n$ and is consistent with the computational evidence described in the source. Its general validity for the stated ranges remains open.