Existence of generalized bent functions in the exceptional $q\equiv2\pmod4$, odd-dimensional case

Abstract

We resolve an open problem of Kumar, Scholtz, and Welch (1985) by constructing generalized bent functions from $(\mathbb{Z}/q\mathbb{Z})^m$ to $\mathbb{Z}/q\mathbb{Z}$ in the exceptional case $q\equiv2\pmod4$ with $m$ odd, the case their paper left without a construction and which four decades of subsequent work had addressed only through nonexistence results. Concretely, for all integers $m\geq d\geq 2$ with $m=dr+2k$, $r\geq 1$, $k\geq 0$, we construct an explicit generalized bent function from $(\mathbb{Z}/q\mathbb{Z})^m$ to $\mathbb{Z}/q\mathbb{Z}$, where $q=2(2^d-1)$. We further show that, when $d\ge3$, these generalized bent functions have Fourier coefficients that are not roots of unity --- all of them when $r$ is odd --- which gives a negative answer to a recent question of Armario, Egan, Kharaghani, and \'O~Cath\'ain about bent vectors for character tables.