Maximal sets of mutually orthogonal frequency squares

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