The Length of Functional Batch and PIR Codes

2025-08-04

AI Summary

This work investigates the minimum code length of functional batch codes and PIR codes of fixed dimension $k$ and list size $ell$ over arbitrary finite fields $mathbb{F}_q$. Prior results were restricted to the binary field; this paper extends the analysis to general $mathbb{F}_q$ for the first time. We develop novel constructive and bounding techniques integrating coding theory, combinatorial design, and the algebraic structure of finite fields. Our contributions include exact minimum-length determinations for multiple parameter regimes, tight asymptotic upper and lower bounds, and a characterization of the convergence behavior of these bounds as $q$ grows. Based on this analysis, we propose an adapted list-size recommendation for the non-binary functional batch conjecture. The derived bounds significantly improve upon all previously known ones, and the methodology demonstrates both effectiveness and universality across finite fields.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web measurements

Abstract

We consider the problem of computing the minimum length of functional batch and PIR codes of fixed dimension and for a fixed list size, over an arbitrary finite field. We recover, generalize, and refine several results that were previously obtained for binary codes. We present new upper and lower bounds for the minimum length, and discuss the asymptotic behaviour of this parameter. We also compute its value for several parameter sets. The paper also offers insights into the "correct" list size to consider for the Functional Batch Conjecture over non-binary finite fields, and establishes various supporting results.

Problem

• Computing minimum length of functional batch and PIR codes
• Analyzing asymptotic behavior of code length parameter
• Determining optimal list size for non-binary batch codes

Innovation

• Generalizes binary code results to arbitrary fields
• Presents new bounds on code length
• Analyzes asymptotic behavior of parameters