{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:02Z","timestamp":1753893842730,"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>A frequency square is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only binary frequency squares of order $n$ with $n\/2$ zeros and $n\/2$ ones in each row and column. Two such frequency squares are orthogonal if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a set of $k$-MOFS$(n)$ is a set of $k$ binary frequency squares of order $n$ in which each pair of squares is orthogonal.\r\nA set of $k$-MOFS$(n)$ must satisfy $k\\le(n-1)^2$, and any set of MOFS achieving this bound is said to be complete. For any $n$ for which there exists a Hadamard matrix of order $n$ we show that there exists at least $2^{n^2\/4-O(n\\log n)}$ isomorphism classes of complete sets of MOFS$(n)$. For $2&lt;n\\equiv2\\pmod4$ we show that there exists a set of $17$-MOFS$(n)$ but no complete set of MOFS$(n)$.\r\nA set of $k$-maxMOFS$(n)$ is a set of $k$-MOFS$(n)$ that is not contained in any set of $(k+1)$-MOFS$(n)$. By computer enumeration, we establish that there exists a set of $k$-maxMOFS$(6)$ if and only if $k\\in\\{1,17\\}$ or $5\\le k\\le 15$. We show that up to isomorphism there is a unique $1$-maxMOFS$(n)$ if $n\\equiv2\\pmod4$, whereas no $1$-maxMOFS$(n)$ exists for $n\\equiv0\\pmod4$. We also prove that there exists a set of $5$-maxMOFS$(n)$ for each order $n\\equiv 2\\pmod{4}$ where $n\\geq 6$.<\/jats:p>","DOI":"10.37236\/9373","type":"journal-article","created":{"date-parts":[[2020,7,9]],"date-time":"2020-07-09T01:56:21Z","timestamp":1594259781000},"source":"Crossref","is-referenced-by-count":2,"title":["Mutually Orthogonal Binary Frequency Squares"],"prefix":"10.37236","volume":"27","author":[{"given":"Thomas","family":"Britz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicholas J.","family":"Cavenagh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adam","family":"Mammoliti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ian M.","family":"Wanless","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,7,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p7\/8124","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p7\/8124","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,7,9]],"date-time":"2020-07-09T01:56:21Z","timestamp":1594259781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i3p7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,10]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,7,9]]}},"URL":"https:\/\/doi.org\/10.37236\/9373","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,7,10]]},"article-number":"P3.7"}}