Some New Results On Orthogonal Frequency Squares

Let n =λm. An F(n; λ) frequency square is an n × n array in which each of m distinct symbols appears exactly A times in each row and column. Two such squares are said to be orthogonal if upon superposition, each of the m2 distinct ordered pairs occurs exactly λ2 times. Hedayat, Sloane and Stufken in their book << Orthogonal Arrays: Theory and Applications >> provided a small table of lower bounds for the maximum number of mutually orthogonal frequency squares of type F(n; λ) (HSS Table), and posed the following research problem (Research Problem 8.22 in their book).Problem Improve the lower bound for the entries in HSS Table, and extend the table.Very recently, Laywine and Mullen extended the table (L-M Table). In this article we will give some construction methods of mutually orthogonal frequency squares. As an application we improve the lower bounds for almost half of the entries in L-M Table and conclude a new table.