Mutually Orthogonal Binary Frequency Squares of Mixed Type

A frequency square of type (n; λ0, λ1,...,λm−1) is an n × n array with entriesfrom the set {0, 1,...,m − 1}, where entry i occurs λi times in every row and λitimes in every column. We say that symboli has frequency λi and note that n = m−1i=0 λi. Let F1 and F2 be frequency squares of type (n; λ0, λ1,...,λm1−1) and(n;μ0, μ1,...,μm2−1), respectively. Then, F1 and F2 are mutually orthogonal if,when superimposed, each of the m1m2 possible ordered pairs(i, j ) with 0 ⩽ i<m1and 0 ⩽ jmutually orthogonal if every two distinct members of the set are orthogonal. A setof k mutually orthogonal frequency squares (MOFS) of order n is said to be a setof k-MOFS(n), or simply a set of k-MOFS. If we do not require that every squarein a set of MOFS has the same type, then we say that the MOFS have mixed type.Since the unique frequency square of type (n; n) is trivially orthogonal to everyother frequency square, we assume that all our frequency squares have at least twosymbols.</m1