{"id":548754,"date":"2018-11-07T14:02:19","date_gmt":"2018-11-07T22:02:19","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=548754"},"modified":"2018-11-07T14:16:33","modified_gmt":"2018-11-07T22:16:33","slug":"lower-bounds-for-2-query-lccs-over-large-alphabet","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/lower-bounds-for-2-query-lccs-over-large-alphabet\/","title":{"rendered":"Lower bounds for 2-query LCCs over large alphabet"},"content":{"rendered":"

A locally correctable code (LCC) is an error correcting code that allows
\ncorrection of any arbitrary coordinate of a corrupted codeword by querying only
\na few coordinates. We show that any {\\em zero-error} $2$-query locally
\ncorrectable code $\\mathcal{C}: \\{0,1\\}^k \\to \u03a3^n$ that can correct a
\nconstant fraction of corrupted symbols must have $n \\geq \\exp(k\/\\log|\u03a3|)$.
\nWe say that an LCC is zero-error if there exists a non-adaptive corrector
\nalgorithm that succeeds with probability $1$ when the input is an uncorrupted
\ncodeword. All known constructions of LCCs are zero-error.<\/p>\n

Our result is tight upto constant factors in the exponent. The only previous
\nlower bound on the length of 2-query LCCs over large alphabet was
\n$\u03a9\\left((k\/\\log|\u03a3|)^2\\right)$ due to Katz and Trevisan (STOC 2000).
\nOur bound implies that zero-error LCCs cannot yield $2$-server private
\ninformation retrieval (PIR) schemes with sub-polynomial communication. Since
\nthere exists a $2$-server PIR scheme with sub-polynomial communication (STOC
\n2015) based on a zero-error $2$-query locally decodable code (LDC), we also
\nobtain a separation between LDCs and LCCs over large alphabet.<\/p>\n

For our proof of the result, we need a new decomposition lemma for directed
\ngraphs that may be of independent interest. Given a dense directed graph $G$,
\nour decomposition uses the directed version of Szemer\u00e9di regularity lemma due
\nto Alon and Shapira (STOC 2003) to partition almost all of $G$ into a constant
\nnumber of subgraphs which are either edge-expanding or empty.<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any {\\em zero-error} $2$-query locally correctable code $\\mathcal{C}: \\{0,1\\}^k \\to \u03a3^n$ that can correct a constant fraction of corrupted symbols must have $n \\geq \\exp(k\/\\log|\u03a3|)$. […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-548754","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2017-1-1","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":0,"msr_main_download":"","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"doi","viewUrl":"false","id":"false","title":"10.4230\/LIPIcs.APPROX-RANDOM.2017.30","label_id":"243109","label":0}],"msr_related_uploader":"","msr_attachments":[],"msr-author-ordering":[{"type":"user_nicename","value":"Sivakanth 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