Geodesic
Antipodal $n$-angles inscribed into the regular $(2n-1)$-angle and half-circulant Hadamard matrices of order $4n$
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 1, pp. 45-66 Cet article a éte moissonné depuis la source Math-Net.Ru

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It's discovered a necessary and sufficient conditions for existence the half-circulant Hadamard matrix of order $4n$ and besides in two forms — geometric and analitic ones. The geometrical necessary and sufficient conditions are being reduced to a question of existence antipodal $n$-angles inscribed into the regular $(2n-1)$-angle while the analitical one — to solvability in the field of real numbers a nonhomogeneous system square $5n-3$ equations with $4n-4$ unknown quantities, which closely connect with a some cubic nonreducible smoth hypersurface in $(2n-1)$-dimensional projective space. 
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     author = {A. I. Medianik},
     title = {Antipodal $n$-angles inscribed into the regular $(2n-1)$-angle and half-circulant {Hadamard} matrices of order~$4n$},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {45--66},
     year = {2004},
     volume = {11},
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     url = {http://geodesic.mathdoc.fr/item/JMAG_2004_11_1_a2/}
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A. I. Medianik. Antipodal $n$-angles inscribed into the regular $(2n-1)$-angle and half-circulant Hadamard matrices of order $4n$. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004) no. 1, pp. 45-66. http://geodesic.mathdoc.fr/item/JMAG_2004_11_1_a2/