{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article1938","name":"Uniqueness of Optimal Mod 3 Circuits for Parity","abstract":"We prove that the quadratic polynomials modulo $3$\r\n with the largest correlation with parity are unique up to\r\n permutation of variables and constant factors. As a consequence of\r\n our result, we completely characterize the smallest \r\nMAJ~$circ mbox{MOD}_3 circ {\r\nm AND}_2$ circuits that compute parity, where a\r\n MAJ~$circ mbox{MOD}_3 circ {\r\nm AND}_2$ circuit is one that has a\r\n majority gate as output, a middle layer of MOD$_3$ gates and a\r\n bottom layer of AND gates of fan-in $2$. We\r\n also prove that the sub-optimal circuits exhibit a stepped behavior:\r\n any sub-optimal circuits of this class that compute parity \r\n must have size at least a factor of $frac{2}{sqrt{3}}$ times the\r\n optimal size. This verifies, for the special case of $m=3$,\r\n two conjectures made\r\n by Due~{n}ez, Miller, Roy and Straubing (Journal of Number Theory, 2006) for general MAJ~$circ mathrm{MOD}_m circ\r\n {\r\nm AND}_2$ circuits for any odd $m$. The correlation\r\n and circuit bounds are obtained by studying the associated\r\n exponential sums, based on some of the techniques developed \r\n by Green (JCSS, 2004). We regard this as a step towards\r\n obtaining tighter bounds both for the $m \r\not = 3$ quadratic\r\n case as well as for\r\n higher degrees.","keywords":["Circuit complexity","correlations","exponential sums"],"author":[{"@type":"Person","name":"Green, Frederic","givenName":"Frederic","familyName":"Green"},{"@type":"Person","name":"Roy, Amitabha","givenName":"Amitabha","familyName":"Roy"}],"position":7,"pageStart":1,"pageEnd":15,"dateCreated":"2008-01-15","datePublished":"2008-01-15","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Green, Frederic","givenName":"Frederic","familyName":"Green"},{"@type":"Person","name":"Roy, Amitabha","givenName":"Amitabha","familyName":"Roy"}],"copyrightYear":"2008","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07411.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume669","volumeNumber":7411,"name":"Dagstuhl Seminar Proceedings, Volume 7411","dateCreated":"2008-01-15","datePublished":"2008-01-15","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article1938","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume669"}}}