Query-Efficient Locally Decodable Codes of Subexponential Length

TL;DR

This paper advances the construction of query-efficient locally decodable codes (LDCs) with subexponential length by exploring algebraic properties of certain integers, leading to improved query complexity and new PIR schemes.

Contribution

It identifies algebraic properties of specific integers to construct more query-efficient LDCs, improving upon previous query complexities and enabling new private information retrieval schemes.

Findings

Constructed LDCs with fewer queries than prior work.

Identified 50 special integers with algebraic properties for LDC construction.

Developed new PIR schemes based on the improved LDCs.

Abstract

We develop the algebraic theory behind the constructions of Yekhanin (2008) and Efremenko (2009), in an attempt to understand the ``algebraic niceness'' phenomenon in $\mathbb{Z}_m$. We show that every integer $m = pq = 2^t -1$, where $p$, $q$ and $t$ are prime, possesses the same good algebraic property as $m=511$ that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki's composition method. More precisely, we construct a $3^{\lceil r/2\rceil}$-query LDC for every positive integer $r<104$ and a $\left\lfloor (3/4)^{51}\cdot 2^{r}\right\rfloor$-query LDC for every integer $r\geq 104$, both of length $N_{r}$, improving the $2^r$ queries used by Efremenko (2009) and $3\cdot 2^{r-2}$ queries used by Itoh and Suzuki (2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.