SIAM Journal on Computing, Volume 49, Issue 2, Page 284-317, January 2020.
A robust positioning pattern is a large array that allows a mobile device to locate its position by reading a possibly corrupted small window around it. In this paper, we provide constructions of binary positioning patterns, equipped with efficient locating algorithms, that are robust to a constant number of errors and have redundancy within a constant factor of optimality. Furthermore, we modify our constructions to correct rank errors and obtain binary positioning patterns robust to any errors of rank less than a constant number. Additionally, we construct $q$-ary robust positioning sequences robust to a large number of errors, some of which have length attaining the upper bound. Our construction of binary positioning sequences that are robust to a constant number of errors has the least known redundancy among those explicit constructions with efficient locating algorithms. On the other hand, for binary robust positioning arrays, our construction is the first explicit construction whose redundancy is within a constant factor of optimality. The locating algorithms accompanying both constructions run in time cubic in sequence length or array dimension.

中文翻译：

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低冗余度的稳健定位模式

SIAM计算杂志，第49卷，第2期，第284-317页，2020年1月。
稳健的定位模式是一个大型阵列，允许移动设备通过读取其周围可能损坏的小窗口来定位其位置。在本文中，我们提供了带有有效定位算法的二进制定位模式的构造，该算法对恒定数量的错误具有鲁棒性，并且在恒定的最优因子内具有冗余度。此外，我们修改了结构以纠正秩误差，并获得了对任何小于常数的秩误差均具有鲁棒性的二进制定位模式。此外，我们构造了对大量错误具有鲁棒性的$ q $ ary鲁棒性定位序列，其中一些具有达到上限的长度。在具有有效定位算法的那些显式构造中，我们的对一定数量错误具有鲁棒性的二进制定位序列的构造在冗余度方面鲜为人知。另一方面，对于二元鲁棒定位阵列，我们的构造是第一个显式构造，其冗余度在恒定的最优因子内。伴随两种构造的定位算法在序列长度或数组维度上按时间立方运行。