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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; 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 kMOFS(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 kmaxMOFS(n; λ) is a set of kMOFS(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 kmaxMOFS(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= p^{u} be a prime power and let p^{v} 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 (qh1)2/(q1)maxMOFS(n; n/q).
Original language  English 

Pages (fromto)  525–558 
Number of pages  34 
Journal  Designs Codes and Cryptography 
Volume  89 
DOIs  
Publication status  Published  16 Jan 2021 
Keywords
 Frequency square
 Integral polytope
 MOFS
 Relation
Projects
 1 Finished

Matchings in Combinatorial Structures
Wanless, I., Bryant, D. & Horsley, D.
Australian Research Council (ARC), Monash University, University of Queensland , University of Melbourne
1/01/15 → 10/10/20
Project: Research