The fractional Poisson field (fPf) can be interpreted in term of the number of balls falling down on each point of $\R^D$, when the centers and the radii of the balls are thrown at random following a Poisson point process in $\R^D\times \R^+$ with an appropriate intensity measure. It provides a simple description for a non Gaussian random field that has the same covariance function as the fractional Brownian field. In the present paper, we concentrate on the restrictions of the fPf to finite sets of points in $\R^D$. Actually, since it takes discrete values, it seems natural to adapt this field to a discrete context. We are particularly interested in its finite-dimensional distributions, in its representation on a finite grid, and in its discrete variations which yield an estimator for its Hurst index.