4 problems

Cyclicity conjecture for Abelian planar difference sets. The period of $\mathcal L$ in $A_n$ along $\mathbf f_{i,j}$ is equal to $n^2+n+1$ for at least one vector $\mathbf f_{i,j}$…

Prime power conjecture. There exists a linear $1$-perfect code in $(A_n,d)$, equivalently in $(\mathbb{Z}^n,d_a)$, if and only if $n$ is a prime power.

Let $D_n$ be the classical root lattice of type $D_n$, and let $h_{D_n}(x)$ denote its coordinator polynomial. Conway and Sloane's conjecture. With Colin Mallows's help, Conway and…

Let $A_n$ be the root lattice, and let $L$ be a full-rank sub-lattice of $A_n$ satisfying the uniformity and reflection-invariance properties: its min-genus and max-genus are equal…