  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.
