Regular simplex inscribed into a cube, half-circulant Hadamard matrices and Gaussian sums

It is proved with help of trigonometric sums method that if a number $2n-1$ is prime or equal to product of two prime twin numbers then a half-circulant Hadamard matrix of order $4n$ exists and into a $(4n-1)$-cube one can inscribe a regular simplex of the same dimensions. Group properties of polinomial pairs which give Hadamard matrices of half-circulant type is investigated as well, and it's installed effective necessary existence conditions for a given polinomial (from a group ring over whole numbers) another polinomial which forms with it such the pair what makes use for practical construction of Hadamard matrices of all orders $4n\leq 80$ with help of PC.