Serving Every Symbol: All-Symbol PIR and Batch Codes
Avital Boruchovsky, Anina Gruica, Jonathan Niemann, Eitan Yaakobi

TL;DR
This paper introduces a unified framework for all-symbol PIR and batch codes, determining optimal code lengths, structural properties, and trade-offs, while connecting classical codes like MDS and simplex codes.
Contribution
It unifies various code families under a common framework, characterizes optimal codes, and explores classical codes within this new context.
Findings
Determined minimum code length for small k and t values.
Characterized structural properties of optimal codes.
Established new cases related to the simplex code and an open conjecture.
Abstract
A -all-symbol PIR code and a -all-symbol batch code of dimension consist of servers storing linear combinations of information symbols with the following recovery property: any symbol stored by a server can be recovered from pairwise disjoint subsets of servers. In the batch setting, we further require that any multiset of size of stored symbols can be recovered from~ 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 and , characterize structural properties of codes attaining this optimum, and derive bounds that show the trade-offs between length, dimension, minimum distance, and . In addition, we study MDS codes and the…
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