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

We resolve an open problem of Kumar, Scholtz, and Welch (1985) by constructing generalized bent functions from $(\mathbb{Z}/q\mathbb{Z})^d$ to $\mathbb{Z}/q\mathbb{Z}$ in the exceptional case $q\equiv2\pmod4$ with $d$ odd, the case their paper left without a construction and which four decades of subsequent work had addressed only through nonexistence results. Concretely, for every odd integer $d\geq3$ such that $p=2^d-1$ is a Mersenne prime, we construct an explicit generalized bent function from $(\mathbb{Z}/2p\mathbb{Z})^d$ to $\mathbb{Z}/2p\mathbb{Z}$. In particular, this produces a function of type $[3,14]$. We further show that the Fourier coefficients of these generalized bent functions can not be a root of unity, which gives a negative answer to a recent question of Armario, Egan, Kharaghani, and \'O~Cath\'ain about bent vectors for character tables.