某些图的最优阻力反码
Optimal Tristance Anticodes in Certain Graphs
摘要：
对于$z_1,z_2,z_3 \in \Z^n$，\emph{tristance} $d_3(z_1,z_2,z_3)$是将$\Z^n$上的$L_1$ -距离推广为反映三个点而不是两个点的相对离散度的量。直径为$d$的阻反码$\cA_d$是$\Z^n$的一个子集，其性质是对所有$z_1,z_2,z_3 \in \cA_d$都有$d_3(z_1,z_2,z_3) \leq d$。一个反码是最佳的，如果它有最大可能的基数为直径$d$。我们确定的基数，并完全分类的最佳阻反码在$\Z^2$的所有直径$d \ge 1$。然后，我们将这一结果推广到两个相关的距离模型：$\Z^2$上的不同距离结构，其中$d(z_1,z_2) = 1$，如果$z_1,z_2$是相邻的水平，垂直，或对角，和距离结构时，$\Z^2$被取代的六方晶格$A_2$。我们还研究了$\Z^3$中的最佳tristance反码和$\Z^2$中的最佳quadristance反码，并给出了它们的基数的界。最后，我们简要讨论了我们的结果的应用程序，多维交织方案和连接轨迹在游戏中的围棋。
英文摘要：
For $z_1,z_2,z_3 \in \Z^n$, the \emph{tristance} $d_3(z_1,z_2,z_3)$ is a generalization of the $L_1$-distance on $\Z^n$ to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticode $\cA_d$ of diameter $d$ is a subset of $\Z^n$ with the property that $d_3(z_1,z_2,z_3) \leq d$ for all $z_1,z_2,z_3 \in \cA_d$. An anticode is optimal if it has the largest possible cardinality for its diameter $d$. We determine the cardinality and completely classify the optimal tristance anticodes in $\Z^2$ for all diameters $d \ge 1$. We then generalize this result to two related distance models: a different distance structure on $\Z^2$ where $d(z_1,z_2) = 1$ if $z_1,z_2$ are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when $\Z^2$ is replaced by the hexagonal lattice $A_2$. We also investigate optimal tristance anticodes in $\Z^3$ and optimal quadristance anticodes in $\Z^2$, and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.
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