Private information retrieval (PIR) codes and batch codes are two important types of codes that are designed for coded distributed storage systems and private information retrieval protocols. These codes have been the focus of much attention in recent years, as they enable efficient and secure storage and retrieval of data in distributed systems. In this paper, we introduce a new class of codes called \emph{$(s,t)$-batch codes}. These codes are a type of storage codes that can handle any multi-set of $t$ requests, comprised of $s$ distinct information symbols. Importantly, PIR codes and batch codes are special cases of $(s,t)$-batch codes. The main goal of this paper is to explore the relationship between the number of redundancy symbols and the $(s,t)$-batch code property. Specifically, we establish a lower bound on the number of redundancy symbols required and present several constructions of $(s,t)$-batch codes. Furthermore, we extend this property to the case where each request is a linear combination of information symbols, which we refer to as \emph{functional $(s,t)$-batch codes}. Specifically, we demonstrate that simplex codes are asymptotically optimal functional $(s,t)$-batch codes, in terms of the number of redundancy symbols required, under certain parameter regime.

翻译：私有信息检索码和批处理码是专为编码分布式存储系统及私有信息检索协议设计的两种重要码型。近年来，这些码型因能实现分布式系统中数据的高效安全存储与检索而备受关注。本文提出一类新型码型——$(s,t)$-批处理码。此类存储码能处理由$t$个请求构成的任意多重集，其中包含$s$个不同的信息符号。值得注意的是，私有信息检索码与批处理码均为$(s,t)$-批处理码的特例。本文主要目标在于探究冗余符号数量与$(s,t)$-批处理码性质之间的关系，具体包括：推导所需的冗余符号数量下界，并给出若干$(s,t)$-批处理码的构造方法。此外，我们将该性质扩展至每个请求均为信息符号线性组合的情形，称之为功能性$(s,t)$-批处理码。特别地，我们证明在特定参数条件下，单纯码在所需冗余符号数量方面是渐近最优的功能性$(s,t)$-批处理码。