AI 中文总结
研究根格\(A_3\)中两个极值问题，即等径问题与线性直径完美码的（非）存在性。通过证明相关结论，给出\(A_3\)中直径为\(D\)子集的最大基数公式，得到三维中格直径完美码存在情况的完整分类，还提出关于任意维度最优反码的猜想。
AI 中文摘要
根格\(A_n\)配备图距离（等同于环境\(\ell_1\)度量的一半）与具有非对称曼哈顿度量的\(\mathbb{Z}^n\)等距。我们研究此空间中的两个极值问题：等径问题，即确定最大反码基数；线性直径完美码的（非）存在性，即通过最优反码进行格平铺，在三维中解决了它们。我们表明，对于每个整数\(D\geq0\)，\(A_3\)中直径为\(D\)的子集的最大基数是\(\binom{D + 3}{3}+(D + 1)\lfloor D^2/4\rfloor\)，且该值由两个离散单纯形的平衡差达到。然后我们证明了一个整性细化的单纯形填充障碍：\(\mathbb{Z}^n\)中不对称曼哈顿距离大于\(D\)的子格在\(\mathbb{R}^n\)中诱导出由\((D + 1)\Delta_n\)进行的格填充。将此观察结果与四面体的精确格填充密度相结合，在三维中得到了完整分类：\(A_3\)中的格直径完美码恰好存在于\(D = 1\)和\(D = 2\)时。我们还给出了基数为四的完美\(B_h\)集的等价陈述。最后，我们提出了一个关于任意维度最优反码的猜想，并将其重述为均匀多重集的交集问题。
英文摘要
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 $\binom{D+3}{3}+(D+1)\lfloor D^2/4\rfloor$, 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 $(D+1)Δ_n$ 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.
