Serving Every Symbol: All-Symbol PIR and Batch Codes
Avital Boruchovsky, Anina Gruica, Jonathan Niemann, Eitan Yaakobi
TL;DR
This work extends the theory of private information retrieval and batch codes by introducing all-symbol variants, ASP and ASB, which require recoverability for every codeword symbol. It develops a uniform generator-matrix framework, derives foundational bounds and invariants, and determines minimum lengths for small values of $t$ and fixed $k$, including exact results for $t\in\{3,4\}$ in many cases. The authors connect these new codes to classical families such as MDS and simplex codes, proving dual-distance based limits and confirming several conjectures about simplex codes as $t$-functional batch codes. The results offer both structural insights and practical guidance on how to balance length, dimension, distance, and recovery strength, with several open questions and directions for future work highlighted, including alphabet-size effects and broader code-family applications.
Abstract
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.