为每个符号服务：所有符号PIR和批量码
Serving Every Symbol: All-Symbol PIR and Batch Codes
AI总结：
本文研究了t-所有符号PIR码和批量码，探讨了其最小长度、结构特性及长度、维度、最小距离和t之间的权衡。
摘要：
一个t-所有符号PIR码和一个维度为k的t-所有符号批量码由n个服务器存储k个信息符号的线性组合构成，具有以下恢复性质：任何服务器存储的符号可以从t个不相交的服务器子集恢复。在批量设置中，我们进一步要求任何大小为t的存储符号多集合可以从t个不相交的服务器子集恢复。该框架统一并扩展了若干已知的码族，包括一步多数逻辑可解码码、(功能)PIR码和(功能)批量码。本文确定了某些小k和t值的最小码长，刻画了达到此最优的码的结构特性，并推导了显示长度、维度、最小距离和t之间权衡的界限。此外，我们研究了MDS码和简单码，展示了这些经典族如何融入我们的框架，并建立了关于简单码作为t-功能批量码的最小t的新情况，这涉及YAAKOBI2020中提出的开放猜想。
英文摘要：
A $t$-all-symbol PIR code and a $t$-all-symbol batch code of dimension $k$ consist of $n$ servers storing linear combinations of $k$ information symbols with the following recovery property: any symbol stored by a server can be recovered from $t$ pairwise disjoint subsets of servers. In the batch setting, we further require that any multiset of size $t$ of stored symbols can be recovered from~$t$ disjoint subsets of servers. This framework unifies and extends several well-known code families, including one-step majority-logic decodable codes, (functional) PIR codes, and (functional) batch codes. In this paper, we determine the minimum code length for some small values of $k$ and $t$, characterize structural properties of codes attaining this optimum, and derive bounds that show the trade-offs between length, dimension, minimum distance, and $t$. In addition, we study MDS codes and the simplex code, demonstrating how these classical families fit within our framework, and establish new cases of an open conjecture from \cite{YAAKOBI2020} concerning the minimal $t$ for which the simplex code is a $t$-functional batch code.
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。
