{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:48Z","timestamp":1753893828961,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $B_p$ be the Latin square given by the addition table for the integers modulo an odd prime $p$. Here we consider the properties of Latin trades in $B_p$ which preserve orthogonality with one of the $p-1$ MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in $p$ for the number of times each symbol occurs in such a trade, with an overall lower bound of $(\\log{p})^2\/\\log\\log{p}$ for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in $B_p$ hits the main diagonal either $p$ or at most $p-\\log_2{p}-1$ times. Finally, if $p\\equiv 1\\pmod{6}$ we show the existence of a Latin square which is orthogonal to $B_p$ and which contains a $2\\times 2$ subsquare.<\/jats:p>","DOI":"10.37236\/6338","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:51:06Z","timestamp":1578671466000},"source":"Crossref","is-referenced-by-count":2,"title":["Orthogonal Trades in Complete Sets of MOLS"],"prefix":"10.37236","volume":"24","author":[{"given":"Nicholas","family":"Cavenagh","sequence":"first","affiliation":[]},{"given":"Diane","family":"Donovan","sequence":"additional","affiliation":[]},{"given":"Fatih","family":"Demirkale","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,7,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p15\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p15\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:51:03Z","timestamp":1579236663000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p15"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,28]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6338","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,7,28]]},"article-number":"P3.15"}}