The Length of Functional Batch and PIR Codes

TL;DR

This paper investigates the minimum length of functional batch and PIR codes over finite fields, providing new bounds, asymptotic analysis, and insights into the list size for the Functional Batch Conjecture.

Contribution

It generalizes previous binary code results to arbitrary finite fields and offers new bounds and asymptotic behavior analysis for code length.

Findings

New upper and lower bounds for code length

Asymptotic behavior of the minimum length analyzed

Computed values for specific parameter sets

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.