The $v \times v $$(0,1,-1)$-Circulant Equation $AA^T = v I - J$
James H. McKay, Stuart Sui-Sheng Wang · SIAM Journal on Algebraic and Discrete Methods · 1981
An electromechanical pulse generator has been proposed (J. P. Craig and R. Saeks, An electromechanical pulse generator, Proc. 1st IEEE International Pulsed Power Conference, Institute of Electrical and Electronics Engineers, 1976, pp. IIB 7-1–IIB 7-4) which is equivalent to finding a $v \times v $ circulant matrix A with entries from $\{ 0,1, - 1 \}$ such that $AA^T = v I - J$. In this earlier work it is reported that if v is an odd prime and the entries in the first row of A are the Legendre symbols $(j/v )$, $0\leqq j\leqq v - 1$, then A is a solution. It was conjectured that A exists only if $v $ is an odd prime and that the solution is unique up to cyclic permutation of the columns and multiplication of A by $ - 1$. In this paper, using convolution products, Fourier transforms and number theory, we settle these two conjectures affirmatively.