Good Locally Testable Codes with Small Alphabet and Small Query Size
Uriya First, Stav Lazarovici
TL;DR
The paper completely resolves the existence landscape for good q-query locally testable codes (LTCs) across all alphabets and $\mathbb{F}$-vector spaces, showing that good LTCs exist for all $(q,|\Sigma|)$ except the binary 2-query case. Central to the construction is an alphabet-reduction framework built on concatenation with inner codes (generalized long codes and generalized Hadamard codes) defined by $2$-letter constraints, together with a tester-compatibility/separability toolkit that preserves the LTC property through the transformation. The authors leverage recent results on good $2$-query $\mathbb{F}$-LTCs and develop a general method to reduce alphabet size while maintaining low-query testing, yielding good $2$- and $3$-query LTCs across broad alphabets and vector spaces. This settles a long-standing question in coding theory and property testing and provides a versatile set of tools (alphabet reduction, separable testers) for potential applications in PCPs and related areas. The work also clarifies the limits of 2-query LTCs and highlights the nuanced behavior between linear and nonlinear LTCs under alphabet transformations.
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) $Σ$, the question of whether there is a good $q$-query LTC (resp. $\mathbb{F}$-LTC) with alphabet $Σ$. 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.