置换重构中的局部性编码
Coding for Locality in Reconstructing Permutations
摘要：
以分布式方式存储排列的问题出现在几个常见的场景中，例如大型加密或压缩数据集的有效更新。这个问题可以在组合或编码方法中解决。前一种方法归结为用\textit{locality}表示大的置换集合，即置换的任何符号都可以从其他符号的小集合中计算出来。在后一种方法中，可以对置换进行编码以便实现局部性。本文重点介绍了组合方法。我们提供了一组置换的最大大小的上界和下界的局部性，并提供了几个简单的结构，达到上界。在没有达到上界的情况下，我们提供了替代结构使用里德-所罗门码，置换多项式，和多置换。
英文摘要：
The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding approach. The former approach boils down to presenting large sets of permutations with \textit{locality}, that is, any symbol of the permutation can be computed from a small set of other symbols. In the latter approach, a permutation may be coded in order to achieve locality. This paper focuses on the combinatorial approach. We provide upper and lower bounds for the maximal size of a set of permutations with locality, and provide several simple constructions which attain the upper bound. In cases where the upper bound is not attained, we provide alternative constructions using Reed-Solomon codes, permutation polynomials, and multi-permutations.
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