We give a simple argument which shows that Gabor systems consisting of odd functions of $d$ variables and symplectic lattices of density $2^{d}$ cannot constitute a Gabor frame. In the one-dimensional, separable case, this follows from a more general result of Lyubarskii and Nes [‘Gabor frames with rational density’, Appl. Comput. Harmon. Anal.34(3) (2013), 488–494]. We use a different approach exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.