Good Locally Testable Codes with Small Alphabet and Small Query Size

TL;DR

This paper characterizes the existence of good low-query locally testable codes (LTCs) across various alphabets and linearity conditions, showing that beyond certain small cases, good LTCs are possible for larger alphabets and query sizes.

Contribution

It proves that good 2-query LTCs exist for larger alphabets and 3-query LTCs for binary alphabets, fully resolving the existence question for all query sizes and alphabets.

Findings

Good 2-query LTCs exist on alphabets with more than 2 letters.

Good 3-query LTCs exist with a binary alphabet.

A general method for reducing alphabet size of low-query LTCs is established.

Abstract

Ben-Sasson, Goldreich and Sudan showed that a binary error correcting code admitting a $2$-query tester cannot be good, i.e., it cannot have both linear distance and positive rate. The same holds when the alphabet is a finite field $\mathbb{F}$, the code is $\mathbb{F}$-linear, and the $2$-query tester is $\mathbb{F}$-linear. We show that those are essentially the only limitations on the existence of good locally testable codes (LTCs). That is, there are good $2$-query LTCs on any alphabet with more than $2$ letters, and good $3$-query LTCs with a binary alphabet. Similarly, there are good $3$-query $\mathbb{F}$-linear LTCs, and for every $\mathbb{F}$-vector space $V$ of dimension greater than $1$, there are good $2$-query LTCs with alphabet $V$ whose tester is $\mathbb{F}$-linear. This completely solves, for every $q\geq 2$ and alphabet (resp. $\mathbb{F}$-vector space) $\Sigma$, the question of whether there is a good $q$-query LTC (resp. $\mathbb{F}$-LTC) with alphabet $\Sigma$. Our proof builds on the recent good $2$-query $\mathbb{F}$-LTCs of the first author and Kaufman, by establishing a general method for reducing the alphabet size of a low-query LTC.