On the Modulus in Matching Vector Codes

TL;DR

This paper investigates the properties of good moduli in matching vector codes, demonstrating the existence of infinitely many such numbers and providing methods to generate smaller good numbers with the same prime divisors.

Contribution

It extends the understanding of good moduli in MVCs by proving the existence of infinitely many good numbers and offering explicit construction methods.

Findings

Good numbers of the form p1^α1 p2^α2 are characterized.

Multiples of good numbers of the same form are also good.

Infinitely many new good numbers can be constructed explicitly.

Abstract

A $k$-query locally decodable code (LDC) $C$ allows one to encode any $n$-symbol message $x$ as a codeword $C(x)$ of $N$ symbols such that each symbol of $x$ can be recovered by looking at $k$ symbols of $C(x)$, even if a constant fraction of $C(x)$ have been corrupted. Currently, the best known LDCs are matching vector codes (MVCs). A modulus $m=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}$ may result in an MVC with $k\leq 2^r$ and $N=\exp(\exp(O((\log n)^{1-1/r} (\log\log n)^{1/r})))$. The $m$ is {\em good} if it is possible to have $k<2^r$. The good numbers yield more efficient MVCs. Prior to this work, there are only {\em finitely many} good numbers. All of them were obtained via computer search and have the form $m=p_1p_2$. In this paper, we study good numbers of the form $m=p_1^{\alpha_1}p_2^{\alpha_2}$. We show that if $m=p_1^{\alpha_1}p_2^{\alpha_2}$ is good, then any multiple of $m$ of the form $p_1^{\beta_1}p_2^{\beta_2}$ must be good as well. Given a good number $m=p_1^{\alpha_1}p_2^{\alpha_2}$, we show an explicit method of obtaining smaller good numbers that have the same prime divisors. Our approach yields {\em infinitely many} new good numbers.