{"id":548754,"date":"2018-11-07T14:02:19","date_gmt":"2018-11-07T22:02:19","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;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:\/\/cm-edgetun.pages.dev\/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":"<blockquote class=\"abstract mathjax\"><p>A locally correctable code (LCC) is an error correcting code that allows<br \/>\ncorrection of any arbitrary coordinate of a corrupted codeword by querying only<br \/>\na few coordinates. We show that any {\\em zero-error} $2$-query locally<br \/>\ncorrectable code $\\mathcal{C}: \\{0,1\\}^k \\to \u03a3^n$ that can correct a<br \/>\nconstant fraction of corrupted symbols must have $n \\geq \\exp(k\/\\log|\u03a3|)$.<br \/>\nWe say that an LCC is zero-error if there exists a non-adaptive corrector<br \/>\nalgorithm that succeeds with probability $1$ when the input is an uncorrupted<br \/>\ncodeword. All known constructions of LCCs are zero-error.<\/p>\n<p>Our result is tight upto constant factors in the exponent. The only previous<br \/>\nlower bound on the length of 2-query LCCs over large alphabet was<br \/>\n$\u03a9\\left((k\/\\log|\u03a3|)^2\\right)$ due to Katz and Trevisan (STOC 2000).<br \/>\nOur bound implies that zero-error LCCs cannot yield $2$-server private<br \/>\ninformation retrieval (PIR) schemes with sub-polynomial communication. Since<br \/>\nthere exists a $2$-server PIR scheme with sub-polynomial communication (STOC<br \/>\n2015) based on a zero-error $2$-query locally decodable code (LDC), we also<br \/>\nobtain a separation between LDCs and LCCs over large alphabet.<\/p>\n<p>For our proof of the result, we need a new decomposition lemma for directed<br \/>\ngraphs that may be of independent interest. Given a dense directed graph $G$,<br \/>\nour decomposition uses the directed version of Szemer\u00e9di regularity lemma due<br \/>\nto Alon and Shapira (STOC 2003) to partition almost all of $G$ into a constant<br \/>\nnumber of subgraphs which are either edge-expanding or empty.<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>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 $n \\geq \\exp(k\/\\log|\u03a3|)$. [&hellip;]<\/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":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"RANDOM 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