TITLE: Explicit construction of $n/2$ by $n$ circulant partial Hadamard matrices
QUESTION [5 upvotes]: In Circulant partial Hadamard matrices by Craigen, Faucher, Low, and Wares it is stated in Theorem 9 that there is a $(p+1)$ by $2(p+1)$ circulant partial Hadamard matrix for every prime power $p$. This is very interesting but I would really like an explicit construction and I can't work out if or where one is given.

Is there an explicit construction that gives a $(p+1)$ by $2(p+1)$
  circulant partial Hadamard matrix for prime power $p$?

REPLY [2 votes]: The result you quote is true because negacyclic $C$-matrices of order $p+1$ exist. In On orthogonal matrices, J. Math. and Phys. 12 (1933), Paley gave a construction of $C$-matrices using the Legendre symbol $\chi$ of the Galois field GF$(p)$. A variation of this construction leads to a negacyclic form for these Paley matrices. The reference for these results is Delsarte, Goethals, and Seidel, Orthogonal matrices with zero diagonal. II, Can. J. Math., Vol. XXIII, No. 5 (1971).