On designs of maximal $(+1, -1)$-matrices of order $n\equiv 2(\textrm {mod}\ 4)$. II

Abstract:

Finding maximal $( + 1, - 1)$-matrices ${M_{2m}}$ of order $2m$ (with odd $m$) constructible in the standard form \[ \left ( {\begin {array}{*{20}{c}} A & B \\ { - {B^T}} & {{A^T}} \\ \end {array} } \right )\] is reduced to the finding of two polynomials $C(w)$, $D(w)$(corresponding to the circulant submatrices $A$, $B$) satisfying \begin{equation}\tag {$*$} |C(w)|^2 + |D(w){|^2} = \tfrac {1}{2}(m - 1),\end{equation} , where $w$ is any primitive $m$th root of unity. Thus, all ${M_{2m}}$ constructible by the standard form (see [4]) can be classified by the formula $\left ( * \right )$. Some new matrices ${M_{2m}}$ for $m = 25,27,31$, were found by this method.