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首页> 外文期刊>Information Theory, IEEE Transactions on >Two-Dimensional Patterns With Distinct Differences—Constructions, Bounds, and Maximal Anticodes
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Two-Dimensional Patterns With Distinct Differences—Constructions, Bounds, and Maximal Anticodes

机译：具有明显差异的二维模式-构造，边界和最大反码

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摘要

A two-dimensional (2-D) grid with dots is called a 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 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid.
机译：如果连接两个点的任何两条线的长度或斜率不同，则带有点的二维（2-D）网格称为具有明显差异的配置。已知这些配置具有许多应用，例如雷达，声纳，物理对准和时间位置同步。而不是像以前研究的那样，将点限制在正方形或矩形中，我们限制了配置点之间的最大距离。其动机是这种配置在无线传感器网络中的密钥分发中的新应用。我们考虑六边形网格以及传统的正方形网格中的配置，并以欧几里得度量，曼哈顿度量或六角形度量来测量距离。我们注意到，这些配置被限制在相应网格中的最大反码内。我们对每个网格中每个直径的最大反码进行分类。我们提出了模式中点数的上限，这些最大反码中包含明显的差异。我们的界限（否定）解决了Golomb和Taylor关于任意大尺寸蜂窝阵列的存在的问题。通过考虑正方形网格中具有明显差异的周期性构造，我们提出了在各种二维形状（例如反码）中包含具有明显差异的构造中的点数的构造和下界。

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