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Codes Cryptogr."],"published-print":{"date-parts":[[2020,10]]},"abstract":"Abstract<\/jats:title>Two $$n \\times n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n \u00d7<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> Latin squares $$L_1, L_2$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are said to be orthogonal<\/jats:italic> if, for every ordered pair (x<\/jats:italic>,\u00a0y<\/jats:italic>) of symbols, there are coordinates (i<\/jats:italic>,\u00a0j<\/jats:italic>) such that $$L_1(i,j) = x$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$L_2(i,j) = y$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n y<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A k<\/jats:italic>-MOLS<\/jats:italic> is a sequence of k<\/jats:italic> pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k<\/jats:italic>, log-asymptotically tight bounds on the number of k<\/jats:italic>-MOLS. To study the situation when k<\/jats:italic> grows with n<\/jats:italic>, we bound the number of ways a k<\/jats:italic>-MOLS can be extended to a $$(k+1)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n k<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-MOLS. These bounds are again tight for constant k<\/jats:italic>, and allow us to deduce upper bounds on the total number of k<\/jats:italic>-MOLS for all k<\/jats:italic>. These bounds are close to tight even for k<\/jats:italic> linear in n<\/jats:italic>, and readily generalise to the broader class of gerechte designs, which include Sudoku squares.<\/jats:p>","DOI":"10.1007\/s10623-020-00771-6","type":"journal-article","created":{"date-parts":[[2020,6,30]],"date-time":"2020-06-30T15:03:53Z","timestamp":1593529433000},"page":"2187-2206","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Enumerating extensions of mutually orthogonal Latin squares"],"prefix":"10.1007","volume":"88","author":[{"given":"Simona","family":"Boyadzhiyska","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shagnik","family":"Das","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tibor","family":"Szab\u00f3","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,6,30]]},"reference":[{"issue":"5","key":"771_CR1","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1080\/00029890.2008.11920542","volume":"115","author":"RA Bailey","year":"2008","unstructured":"Bailey R.A., Cameron P.J., Connelly R.: Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes. 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