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.

翻译：一个$t$-全符号PIR码和一个$t$-全符号批量码具有维度$k$，由$n$个服务器组成，存储$k$个信息符号的线性组合，并满足以下恢复性质：任何服务器存储的符号都能从$t$个两两不相交的服务器子集中恢复。在批量场景下，我们进一步要求：任何大小为$t$的存储符号多重集都能从$t$个不相交的服务器子集中恢复。该框架统一并扩展了多个著名码族，包括一步多数逻辑可译码、(函数型)PIR码和(函数型)批量码。本文针对较小的$k$和$t$值确定了最小码长，刻画了达到这一最优值的码的结构性质，并推导了展示长度、维度、最小距离与$t$之间权衡关系的界。此外，我们研究了MDS码和单形码，展示了这些经典码族如何适用于我们的框架，并建立了文献\cite{YAAKOBI2020}中关于单形码成为$t$-函数型批量码的最小$t$值的开放猜想的新情形。