3 problems

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

Interval conjecture. For every dimension $n$, the values $N\in [D(n,\tau),D(n,\tau+1)]$ for which second-level bounds exist form an interval.

Fix a dimension $d\geq 3$, density $\rho=1$, and $\alpha>0$. For the potential $p(r)=e^{-\alpha r^2}$, consider the optimal linear programming bound $\widehat{f}(0)-f(0)$ from the…