An Isodiametric Theorem and Lattice Diameter-Perfect Codes in $A_3$

Abstract

The root lattice $A_n$, equipped with its graph distance (equivalently, one half of the ambient ${\ell }_1$ metric), is isometric to ${\mathbb{Z}}^n$ with the asymmetric Manhattan metric. We study two extremal problems in this space -- the isodiametric problem, i.e., determining the maximum anticode cardinality, and the (non)existence of linear diameter-perfect codes, i.e., lattice tilings by optimal anticodes -- and solve them in dimension $3$. We show that, for every integer $D\ge 0$, the largest cardinality of a diameter-$D$ subset of $A_3$ is $\left(\genfrac{}{}{0ex}{}{D+3}{3}\right)+(D+1)\lfloor D^2/4\rfloor$, and this value is attained by the balanced difference of two discrete simplices. We then prove an integrality-refined simplex-packing obstruction: a sublattice of ${\mathbb{Z}}^n$ of asymmetric Manhattan distance greater than $D$ induces a lattice packing by $(D+1)\hat{I}{”}_n$ in ${\mathbb{R}}^n$. Combining this observation with the exact lattice-packing density of the tetrahedron yields a complete classification in dimension $3$: lattice diameter-perfect codes in $A_3$ exist precisely for $D=1$ and $D=2$. We also give the equivalent statement for perfect $B_h$ sets of cardinality four. Finally, we formulate a conjecture regarding optimal anticodes in arbitrary dimension, and restate it as an intersection problem for uniform multisets.