{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:31Z","timestamp":1753893811115,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>There are many nonisomorphic orthogonal arrays with parameters $OA(s^3,s^2+s+1,s,2)$ although the existence of the arrays yields many restrictions.  We denote this by $OA(3,s)$ for simplicity. V. D. Tonchev showed that for even the case of $s=3$, there are at least 68 nonisomorphic orthogonal arrays.  The arrays that are constructed by the $n-$dimensional finite spaces have parameters $OA(s^n, (s^n-1)\/(s-1),s,2)$.  They are called Rao-Hamming type.  In this paper we characterize the $OA(3,s)$ of 3-dimensional Rao-Hamming type.  We prove several results for a special type of $OA(3,s)$ that satisfies the following condition: For any three rows in the orthogonal array, there exists at least one column, in which the entries of the three rows equal to each other. We call this property $\\alpha$-type. We prove the following. (1) An $OA(3,s)$ of $\\alpha$-type exists if and only if $s$ is a prime power. (2) $OA(3,s)$s of $\\alpha$-type are isomorphic to each other as orthogonal arrays. (3) An $OA(3,s)$ of $\\alpha$-type yields $PG(3,s)$. (4) The 3-dimensional Rao-Hamming is an $OA(3,s)$ of $\\alpha$-type. (5) A linear $OA(3,s)$ is of $\\alpha $-type. <\/jats:p>","DOI":"10.37236\/556","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:52:01Z","timestamp":1578714721000},"source":"Crossref","is-referenced-by-count":0,"title":["Orthogonal Arrays with Parameters $OA(s^3,s^2+s+1,s,2)$ and 3-Dimensional Projective Geometries"],"prefix":"10.37236","volume":"18","author":[{"given":"Kazuaki","family":"Ishii","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,3,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p69\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p69\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:14:55Z","timestamp":1579302895000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p69"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3,31]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/556","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,3,31]]},"article-number":"P69"}}