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Good Locally Testable Codes with Small Alphabet and Small Query Size

Authors Uriya A. First , Stav Lazarovici

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LIPIcs.APPROX-RANDOM.2026.70.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing β†’ Coding theory
  • Theory of computation β†’ Computational complexity and cryptography
Keywords
  • error correcting code
  • locally testable code
  • property testing

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Abstract

Ben-Sasson, Goldreich and Sudan [Ben-Sasson et al., 2003] showed that a binary error correcting code admitting a 2-query tester cannot be good, i.e., it cannot have both linear distance and constant rate. They also showed that there are no good codes if the alphabet is a finite field 𝔽, the code is 𝔽-linear, and the 2-query tester is 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 𝔽-linear LTCs, and for every 𝔽-vector space V of dimension greater than 1, there are good 2-query LTCs with alphabet V whose tester is 𝔽-linear. This completely solves, for every q β‰₯ 2 and alphabet (resp. 𝔽-vector space) Ξ£, the question of whether there is a good q-query LTC (resp. 𝔽-LTC) with alphabet Ξ£. Our proof builds on the recent good 2-query 𝔽-LTCs of the first author and Kaufman [First and Kaufman, 2024], by establishing a general method for reducing the alphabet size of a good low-query LTC.

Cite As Get BibTex

Uriya A. First and Stav Lazarovici. Good Locally Testable Codes with Small Alphabet and Small Query Size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 70:1-70:23, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.70

Author Details

Uriya A. First
  • University of Haifa, Israel
Stav Lazarovici
  • University of Haifa, Israel

Funding

This research was supported by ISF grant no. 721/24.

Acknowledgements

We are grateful to the anonymous referees for many useful suggestions, and to Esty Kelman for finding a mistake in an earlier version of Section 4.

References

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