Serving Every Symbol: All-Symbol PIR and Batch Codes

A $t$-all-symbol PIR code and a $t$-all-symbol batch code of dimension $k$ consist of $n$ servers storing linear combinations of $k$ linearly independent 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.