  The goal of this paper is to explicitly describe a minimal binomial
generating set of a class of lattice ideals, namely the ideal of certain
Veronese $3$-fold projections. More precisely, for any integer $d\ge 4$ and any
$d$-th root $e$ of 1 we denote by $X_d$ the toric variety defined as the image
of the morphism $\varphi _{T_d}:\mathbb{P}^3 \longrightarrow \mathbb{P}^{\mu
(T_d)-1}$ where $T_d$ are all monomials of degree $d$ in $k[x,y,z,t]$ invariant
under the action of the diagonal matrix $M(1,e,e^2,e^3).$ In this work, we
describe a $\mathbb{Z}$-basis of the lattice $L_{\eta }$ associated to $I(X_d)$
as well as a minimal binomial set of generators of the lattice ideal
$I(X_d)=I_+(\eta)$.
