The concept of M-structures is given. On accordance with it Hadamard matrice of order 4t(8t) is considered as blocked matrice of 16 (64) blocks of dimension \(t\times t\). Hadamard matrices of Williamson, Goethals-Seidel and Wallis-Whiteman type are considered as M-structures. For further research M-structures with circulant blocks are distinguished by their perspectivity, and that forms the basis of theorem.
Theorem. Suppose there are T-matrices of order t. Further suppose there is an \(OD(4s;u_ 1,...,u_ n)\) constructed of sixteen circulant (or type 1) \(s\times s\) blocks on the variables \(x_ 1,...,x_ n\). Then there is an \(OD(4st;tu_ 1,...,tu_ n)\). In particular, if there is an OD(4s;s,s,s,s) constructed of sixteen circulant (or type 1) \(s\times s\) blocks then there is an OD(4st;st,st,st,st).
The existence of OD(20t;5t,5t,5t,5t) and OD(36t;9t,9t,9t,9t), where t is the order of T-matrices, is proved. Symmetric matrices of Williamson type of new orders are constructed.