Maximal sets of mutually orthogonal frequency squares

Nicholas J. Cavenagh, Adam Mammoliti, Ian M. Wanless

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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; λ) if it contains n/ λ symbols, each of which occurs λ times per row and λ times per column. In the case when λ= n/ 2 we refer to the frequency square as binary. A set of k-MOFS(n; λ) is a set of k frequency squares of type (n; λ) such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of k-maxMOFS(n; λ) is a set of k-MOFS(n; λ) that is not contained in any set of (k+ 1) -MOFS(n; λ). For even n, let μ(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 μ(n) = 1 if n/2 is odd and μ(n) > 1 if n/2 is even. Extending this result, we show that if n/2 is even, then μ(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⩽ k. In particular, this means that lim sup μ(n) is unbounded. Nevertheless we can construct infinite families of maximal binary MOFS of fixed cardinality. More generally, let q= pu be a prime power and let pv be the highest power of p that divides n. If 0 ⩽ v- uh< u/ 2 for h⩾ 1 then we show that there exists a set of (qh-1)2/(q-1)-maxMOFS(n; n/q).

Original languageEnglish
Pages (from-to)525–558
Number of pages34
JournalDesigns Codes and Cryptography
Publication statusPublished - 16 Jan 2021


  • Frequency square
  • Integral polytope
  • MOFS
  • Relation

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