Semi Williamson Type Matrices and the $W(2n,n)$ Conjecture

Abstract

Four $(1,-1,0)$-matrices of order $m$, $X=(x_{ij})$, $Y=(y_{ij})$, $Z=(z_{ij})$, $U=(u_{ij})$ satisfying

1. $X{X}^T+Y{Y}^T+Z{Z}^T+U{U}^T=2m{I}_m$,
2. $x_{ij}^2+y_{ij}^2+z_{ij}^2+u_{ij}^2=2,i,j=1,\dots ,m$,
3. $X,Y,Z,U$ mutually amicable,

will be called semi Williamson type matrices of order $m$. In this paper, we prove that if there exist Williamson type matrices of order $n_1,\dots ,n_k$, then there exist semi Williamson type matrices of order $N=\prod _{j=1}^k{n}_j^{r_j}$, where $r_j$ are non-negative integers. As an application, we obtain a $W(4N,2N)$.
Although the paper presents no new $W(4n,2n)$ for $n$ odd, $n<3000$, it is a step towards proving the conjecture that there exists a $W(4n,2n)$ for any positive integer $n$. This conjecture is a sub-conjecture of the Seberry conjecture $[4,page92]$ that $W(4n,k)$ exist for all $k=0,1,\dots ,4n$. In addition, we find infinitely many new $W(2n,n)$, $n$ odd and the sum of two squares.