Let X be an N by N matrix of +-1s, where N is not divisible by 4. Hadamard`s maximum determinant problem seeks to find X that maximizes the value of Det(X^TX). Such an X is called a D-optimal design. The current state of knowledge on D-optimal designs is tabulated on the webpage:
http://www.indiana.edu/~maxdet/.
Finding achievable upper bounds for Det(X^TX) is essential in finding D-optimal designs. The best known upper bounds for Det(X^TX) are the Barba bound for N=1 (mod 4), Ehlich/Wojtas bound for N=2 (mod 4), and Ehlich bound for N=3 (mod 4).
Det(X^TX) is the product of eigenvalues of X^TX, where each eigenvalue is an algebraic integer in R. The first part of this research will use the approach of Cheng [1978. Optimality of certain asymmetrical experimental designs. Ann. Stat. 6, 1239-1261] and the properties of algebraic integers to improve the aforementioned best known upper bounds for Det(X^TX).
Let l be an odd integer. Binary Legendre pairs of length l can be used can be used to construct a Hadamard matrix of size 2l+2 Fletcher, Gysin, and Seberry [2001. Application of the discrete Fourier transform to the search for generalized Legendre pairs and Hadamard matrices. Australas. J. Combin. 23, 75-86]. Let
z_1k=u_0+u_1w^k+u_2w^2k+...+u_(l-1)w^(l-1)k, and
z_2k=v_0+v_1w^k+v_2w^2k+...+v_(l-1)w^(l-1)k, where
w is a primitive l`th root of unity and {u_i} and {v_i} are binary sequences of length l. Then binary Legendre pairs exists if and only if the system
|z_1k|^2+|z_2k|^2=(l+1)/2 for k=1,2,...,l
has a solution. The number of solutions to this system of equations appears to grow exponentially Fletcher, Gysin, and Seberry [2001. Application of the discrete Fourier transform to the search for generalized Legendre pairs and Hadamard matrices. Australas. J. Combin. 23, 75-86]. Both z_1k and z_2k are sums of roots of unity in a cyclotomic field extension of Q. The second part of this research will focus on exploring ways of exploiting properties of sums of roots of unity to find binary Legendre pairs for l>=49. In particular, theory of cyclotomic field extensions may be useful in studying this problem.
