具有明显差异的二维模式--构造、边界和最大反码
Two-Dimensional Patterns with Distinct Differences -- Constructions, Bounds, and Maximal Anticodes
摘要：
如果连接两个点的任何两条线的长度或斜率不同，则二维网格称为\emph{configuration with distinct differences}。已知这些配置具有许多应用，诸如雷达、声纳、物理对准和时间-位置同步。而不是限制点躺在一个正方形或矩形，如前所述，我们限制的配置点之间的最大距离，这样做的动机是一个新的应用程序，这样的配置在无线传感器网络中的密钥分配。我们认为配置在六边形网格以及在传统的正方形网格中，与测量的距离都在欧几里德度量，并在曼哈顿或六边形度量。我们注意到，这些配置被限制在相应的网格中的最大反码。我们在每个网格中的每个直径分类最大的反码。我们提出了上界的点的数量在这些最大的反码中所包含的具有明显差异的模式。我们的边界解决（在消极的）一个问题的Golomb和泰勒的存在的蜂窝阵列的任意大的尺寸。我们目前的建设和下界上的点的数量在配置中包含在各种二维形状（如反码）的不同差异，考虑周期性的配置，在正方形网格中的不同差异。
英文摘要：
A two-dimensional grid with dots is called a \emph{configuration with distinct differences} if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various two-dimensional shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid.
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