Optimal possibly nonlinear 3-PIR codes of small size

TL;DR

This paper explores the bounds, constructions, and limitations of small 3-PIR codes, including linear and nonlinear types, providing new constructions and identifying minimal lengths for certain parameters.

Contribution

It generalizes the minimum-distance bound for PIR codes, introduces a new construction for linear PIR codes, and determines minimal lengths for binary 3-PIR codes up to dimension 6.

Findings

New bounds for PIR codes

Construction of some 5-PIR codes using packing designs

Nonlinear 3-PIR codes of length 11 and size 128 are necessary for certain parameters

Abstract

First, we state a generalization of the minimum-distance bound for PIR codes. Then we describe a construction for linear PIR codes using packing designs and use it to construct some new 5-PIR codes. Finally, we show that no encoder (linear or nonlinear) for the binary $r$-th order Hamming code produces a 3-PIR code except when $r=2$. We use these results to determine the smallest length of a binary (possibly nonlinear) 3-PIR code of combinatorial dimension up to~6. A binary 3-PIR code of length 11 and size $2^7$ is necessarily nonlinear, and we pose the existence of such a code as an open problem.