Regular simplex inscribed into a cube and Hadamard matrix of half-circulant type

It's discovered a sufficient condition for existence of the Hadamard matrix of order $4n$ ($n$ – natural number) of half-circulant type, which contains two different circulants of order $2n-1$: right and left one (from here the term). A new method of the Hadamard matrices construction, which is geometrical in point of fact and different from the well-known Williamson method, is received. It's proved as well, that there is the Hadamard matrix of order $2(p+1)$ of half-circulant type, where $p$ is odd prime number, whence it follows, that into $2(p+1)$-dimensional cube one can to inscribe a regular simplex of the same dimension.