Serving Every Symbol: All-Symbol PIR and Batch Codes

2026-01-07

arXiv.org

AI Summary

This work investigates t-all-symbol private information retrieval (PIR) codes and batch codes, wherein every stored symbol can be recovered from t disjoint subsets of servers and arbitrary multisets of t symbols can be retrieved in parallel. By leveraging linear coding theory, combinatorial design, and structural analysis, the paper establishes the first systematic theoretical framework for such codes and uncovers their intrinsic connections to classical structures including MDS codes and simplex codes. The main contributions include deriving fundamental trade-offs among code length, dimension, minimum distance, and the parameter t; characterizing the structure of optimal codes for small parameters; and demonstrating that simplex codes satisfy the functional batch property under broader settings, thereby advancing a related open conjecture.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Information Theory
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• all-symbol PIR codes
• batch codes
• code length
• disjoint recovery
• simplex code

Innovation

• all-symbol PIR codes
• all-symbol batch codes
• disjoint recovery sets
• functional batch codes
• simplex code