Let n=λmn=λm. A frequency square F(n;λ)F(n;λ) is an n×nn×n array in which each of mm distinct symbols appears exactly λλ times in each row and column. Two such squares are said to be orthogonal if upon superposition, each of the m2m2 distinct ordered pairs occurs exactly λ2λ2 times. Hedayat, Sloane and Stufken in their book Hedayat et al. (1999) provided a small table of lower bounds for the maximum number of mutually orthogonal frequency squares of type F(n;λ)F(n;λ) (HSS Table), and posed the following research problem: Improve the lower bound for the entries in HSS Table.#R##N##R##N#Laywine and Mullen in 2001 extended the table (LM Table). In this article we will give some new construction methods of mutually orthogonal frequency squares. As an application we improve the lower bounds for more than half of the entries in LM Table.
['Mingchao Li', 'Yan Zhang', 'Beiliang Du']
Some new results on mutually orthogonal frequency squares