Good Locally Testable Codes with Small Alphabet and Small Query Size
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909968566845440 |
|---|---|
| author | First, Uriya Lazarovici, Stav |
| author_facet | First, Uriya Lazarovici, Stav |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16082 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Good Locally Testable Codes with Small Alphabet and Small Query Size First, Uriya Lazarovici, Stav Computational Complexity 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. |
| title | Good Locally Testable Codes with Small Alphabet and Small Query Size |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2512.16082 |