A note on complex conference matrices

Let ${\mathbb{F}}_q$ be the Galois field of order $q=p^m$, p a prime number and m a positive integer. We prove in this article that for any nontrivial multiplicative character ? of ${\mathbb{F}}_q^{?}$ and for any $b\in {\mathbb{F}}_q^{?}$ we have$\sum _{a\in {\mathbb{F}}_q^{?}}?(a)\stackrel{￣}{?(a+b)}=?1$. Whenever q is odd and ? is the Legendre symbol this formula reduces to the well-known Jacobsthal's formula. A complex conference matrix is a square matrix of order n with zero diagonal and unimodular complex numbers elsewhere such that $C^{?}C=(n?1)I$. Paley used finite fields with odd orders $q=p^m$, p prime and the real Legendre symbol to construct real symmetric conference matrices of orders $q+1$ whenever $q\equiv 1(\mathrm{mod}4)$ and real skew-symmetric conference matrices of orders $q+1$ whenever $q\equiv ?1(\mathrm{mod}4)$. In this article we extend Paley construction to the complex setting. We extend Jacobsthal's formula to all other nontrivial characters to produce a complex symmetric conference matrix of order $q+1$ whenever $q\ge 4$ is any prime power as well as a complex skew-symmetric conference matrix of order $q+1$ whenever q is any odd prime power. These matrices were constructed very recently in connection with harmonic Grassmannian codes, by use of finite fields and the character table of their additive characters. We propose here a new proof of their construction by use of the above generalized formula similarly as was done by Paley in the real case. We also classify, up to equivalence, the complex conference matrices constructed with some nontrivial characters. In particular, we prove that the complex conference matrix constructed with any nontrivial multiplicative character ? and that one constructed with ${?}^{p^k}$ for any integer $k=1,...m?1$ are permutation equivalent. Moreover, we determine the spectrum of any complex conference matrix obtained from this construction.