[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84454-en":3,"doc-seo-84454-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84454,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Good Locally Testable Codes with Small Alphabet and Small Query Size","Good locally testable codes (LTCs) with strong distance and constant rate are constrained by prior impossibility results for 2-query testers, especially over a binary alphabet and under F-linearity assumptions. This work shows these are essentially the only limitations: good q-query LTCs exist exactly under explicit conditions on q and the alphabet size. It also provides corresponding existence for F-LTCs over F-vector spaces, and proves alphabet-reduction constructions that preserve low query complexity.","arXiv :2512 . 16082v2 [ cs .CC] 12 Jul 2026  \nGOOD LOCALLY TESTABLE CODES WITH SMALL ALPHABET  \nAND SMALL QUERY SIZE  \nURIYA A. FIRST∗ AND STAV LAZAROVICI ∗  \nAbstract. Ben-Sasson, Goldreich and Sudan [BSGS03] 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 ≥ 2 and alphabet (resp. F-vector space) Σ, the question of whether there is a good q-query LTC (resp. F-LTC) with alphabet Σ . Our proof builds on the recent good 2-query F-LTCs of the first author and Kaufman [FK24b], by establishing a general method for reducing the alphabet  \nsize of a good low-query LTC.  \n1. Introduction  \nThroughout, Σ is a (finite) alphabet and F is a finite field. We let C ⊆ Σn bean error correcting code, which we think of as ranging in an infinite family of codes with block length tending to infinity. The normalized Hamming distance in Σn is denoted δ (·, ·) . We recall other relevant coding theory terminology in Section 2 .  \nInformally, the code C ⊆ Σn is said to be a locally testable code (LTC) if it admits a randomized algorithm—called a tester — that can estimate with high probability whether a word w ∈ Σn is far from C or not by reading only a constant number of letters from w. Formally, we require the tester for C to read at most q letters from w, accept all words in C, and reject every w ∈ Σn −C with probability at least µ · δ (w, C) for some µ > 0. Here, q and µ are independent of n and w. When we wish to specify the implicit constants q and µ, we will say that C ⊆ Σn is a q-query LTC with soundness µ . Also, if not indicated otherwise, we assume that any tester is non-adaptive, i.e., it decides which positions to read from the input word before reading them.  \nThe notion of an LTC arose in the 1990s from the many works on the developing theories of property testing and probabilistically checkable proofs (PCPs) . It first appeared in print in [FS95, Dfn. 9] in a more lax version, in which the requirement that T rejects every w ∈ Σn with probability at least µ · δ(w, C) was imposed only for words that are Ω(1)-far from C. Such a codes are known as weak LTCs. The definition of LTCs which we use here, known as strong LTCs, originates from [GS06, Dfn. 2.2], which intitiated the systematic study of LTCs as objects of independent interest. See [Gol11] for a more detailed history of how LTCs emerged.  \nSince their introduction, it was not clear whether there are LTCs that are also good codes, i.e., LTCs whose relative distance and rate is bounded away from 0; see  \n∗ University of Haifa  \nE-mail addresses: [uriya.first@gmail.com](uriya.first@gmail.com) , [stavlazar2002@gmail.com](stavlazar2002@gmail.com).  \n2 GOOD LOCALLY TESTABLE CODES  \n[Gol11, Open Problem 2], for example. Indeed, Ben-Sasson, Goldreich and Sudan [BSGS03] showed that there are no good 2-query LTCs on a binary alphabet, and no F-linear 2-query LTCs. (Here, an LTC is said to be F-linear if its alphabet is F and its tester checks F-linear constraints.) Further limitations on 2-query LTCsand 3-query LTCs appeared in [BSV12, KR16] . See also [BSGK+ 10] . Roughly atthe same time, a series of works including [PS94, Din07, BSS06, Vid13, KMRZS17, GKO+ 18] established the existence of ‘almost’ good LTCs (both weak and strong), e.g. , LTCs with rate poly(log n)−1 and constant relative distance, or good codes ","cbCaijrFXL1xwqv8","https://ap.wps.com/l/cbCaijrFXL1xwqv8","pdf",438797,1,23,"English","en",105,"# Introduction\n## Main Results\n## Alphabet Reduction Method\n# Theorems and Consequences","[{\"question\":\"What question does the paper resolve about locally testable codes (LTCs)?\",\"answer\":\"It fully determines, for every q ≥ 2 and every alphabet (and F-vector space), when there exists a good q-query LTC (or F-LTC). The solution specifies the exact conditions on q and the alphabet parameters.\"},{\"question\":\"Why were 2-query LTCs previously thought to be limited?\",\"answer\":\"Ben-Sasson, Goldreich, and Sudan showed impossibility for good 2-query LTCs in certain settings, including binary alphabets and F-linear 2-query LTCs. This paper proves those restrictions are essentially the only ones.\"},{\"question\":\"How does the paper construct good LTCs with smaller alphabets or query complexity?\",\"answer\":\"It introduces an alphabet reduction method for low-query LTCs, building on existing good 2-query F-LTCs and using that reduction to obtain good codes for broader alphabet sizes and query counts.\"}]",1784195718,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"good-locally-testable-codes-with-small-alphabet-and-small-query-size","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/good-locally-testable-codes-with-small-alphabet-and-small-query-size/84454/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What question does the paper resolve about locally testable codes (LTCs)?","Question",{"text":74,"@type":75},"It fully determines, for every q ≥ 2 and every alphabet (and F-vector space), when there exists a good q-query LTC (or F-LTC). The solution specifies the exact conditions on q and the alphabet parameters.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why were 2-query LTCs previously thought to be limited?",{"text":79,"@type":75},"Ben-Sasson, Goldreich, and Sudan showed impossibility for good 2-query LTCs in certain settings, including binary alphabets and F-linear 2-query LTCs. This paper proves those restrictions are essentially the only ones.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the paper construct good LTCs with smaller alphabets or query complexity?",{"text":83,"@type":75},"It introduces an alphabet reduction method for low-query LTCs, building on existing good 2-query F-LTCs and using that reduction to obtain good codes for broader alphabet sizes and query counts.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]