纠错码中多请求的利用
On taking advantage of multiple requests in error correcting codes
摘要：
在纠错码--特别是局部可恢复码（LRC）和局部可解码码（LDC）--中的大多数局部性概念中，解码器试图学习消息的单个符号，而只查看相应码字的几个符号。然而，假设想要恢复消息的r> 1个符号。两个极端是重复单查询算法r次（这是具有可用性的LRC、原始多集批处理代码和PIR代码背后的直觉）或简单地运行全局解码算法来恢复整个事情。在本文中，我们将研究在这两个极端之间会发生什么：在什么值的r停止是一个好主意？为了开始研究这个问题，我们引入了强大的批处理代码，它试图找到r个符号的消息使用m个查询的码字，在存在的擦除。我们专注于r = m的情况，这可以被看作是MDS属性的推广。令人惊讶的是，我们表明，这种概念的地方，重复是最佳的，甚至非常大的值的$r = Ω(k)$。
英文摘要：
In most notions of locality in error correcting codes -- notably locally recoverable codes (LRCs) and locally decodable codes (LDCs) -- a decoder seeks to learn a single symbol of a message while looking at only a few symbols of the corresponding codeword. However, suppose that one wants to recover r > 1 symbols of the message. The two extremes are repeating the single-query algorithm r times (this is the intuition behind LRCs with availability, primitive multiset batch codes, and PIR codes) or simply running a global decoding algorithm to recover the whole thing. In this paper, we investigate what can happen in between these two extremes: at what value of r does repetition stop being a good idea? In order to begin to study this question we introduce robust batch codes, which seek to find r symbols of the message using m queries to the codeword, in the presence of erasures. We focus on the case where r = m, which can be seen as a generalization of the MDS property. Surprisingly, we show that for this notion of locality, repetition is optimal even up to very large values of $r = Ω(k)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
