  Recently an algorithm was given in [Garde & Hyv\"onen, SIAM J. Math. Anal.,
2024] for exact direct reconstruction of any $L^2$ perturbation from linearised
data in the two-dimensional linearised Calder\'on problem. It was a simple
forward substitution method based on a 2D Zernike basis. We now consider the
three-dimensional linearised Calder\'on problem in a ball, and use a 3D Zernike
basis to obtain a method for exact direct reconstruction of any $L^3$
perturbation from linearised data. The method is likewise a forward
substitution, hence making it very efficient to numerically implement.
Moreover, the 3D method only makes use of a relatively small subset of boundary
measurements for exact reconstruction, compared to a full $L^2$ basis of
current densities.
