Column optimal strongly threefold orthogonal matrices in a class index eight Original Research Article

The concept of a strongly threefold orthogonal (STO) matrix is studied in A.S. Hedayat and H. Pesotan [J. Statist. Plann. Inference 15 (1986) 11–17; Linear Algebra Appl. 136 (1990) 1–23]. Let C(R,n) be the class of all STO matrices with R rows and index n. We define Lmax(C(R,n))=L if and only if there is a STO matrix in C(R,n) with L columns and every matrix in C(R,n) has at most L columns. A matrix in C(R,n) is called column optimal if it has Lmax(C(R,n)) columns. The column optimality problem for C(R,n) consists in determining Lmax(C(R,n)) and constructing a column optimal matrix in C(R,n). In this paper we study the column optimality problem for the sequence of classes C(8t+4,8),tgreater-or-equal, slanted3. It is shown that (1) Lmax (C(28,8)) = 6, (2) Lmax(C(16t+12,8))=9 when tgreater-or-equal, slanted46 and (3) 8less-than-or-equals, slantLmax(C(16t+4,8))less-than-or-equals, slant9 when tgreater-or-equal, slanted27. Corresponding column optimal STO matrices in the classes C(28, 8) and C(16t+12,8), tgreater-or-equal, slanted46 are also constructed.