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- W3084403602 abstract "A frequency square is a square matrix in which each row and column is a permutation of the same multiset of symbols. A frequency square is of type $$(n;lambda )$$ if it contains $$n/lambda $$ symbols, each of which occurs $$lambda $$ times per row and $$lambda $$ times per column. In the case when $$lambda =n/2$$ we refer to the frequency square as binary. A set of k-MOFS $$(n;lambda )$$ is a set of k frequency squares of type $$(n;lambda )$$ such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of k-maxMOFS $$(n;lambda )$$ is a set of k-MOFS $$(n;lambda )$$ that is not contained in any set of $$(k+1)$$ -MOFS $$(n;lambda )$$ . For even n, let $$mu (n)$$ be the smallest k such that there exists a set of k-maxMOFS(n; n/2). It was shown in Britz et al. (Electron. J. Combin. 27(3):#P3.7, 26 pp, 2020) that $$mu (n)=1$$ if n/2 is odd and $$mu (n)>1$$ if n/2 is even. Extending this result, we show that if n/2 is even, then $$mu (n)>2$$ . Also, we show that whenever n is divisible by a particular function of k, there does not exist a set of $$k'$$ -maxMOFS(n; n/2) for any $$k'leqslant k$$ . In particular, this means that $$limsup mu (n)$$ is unbounded. Nevertheless we can construct infinite families of maximal binary MOFS of fixed cardinality. More generally, let $$q=p^u$$ be a prime power and let $$p^v$$ be the highest power of p that divides n. If $$0leqslant v-uh<u/2$$ for $$hgeqslant 1$$ then we show that there exists a set of $$(q^h-1)^2/(q-1)$$ -maxMOFS(n; n/q)." @default.
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- W3084403602 date "2021-01-16" @default.
- W3084403602 modified "2023-09-25" @default.
- W3084403602 title "Maximal sets of mutually orthogonal frequency squares" @default.
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- W3084403602 doi "https://doi.org/10.1007/s10623-020-00832-w" @default.
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