We present an algebraic classification approach to Calabi-Yau complete intersections in fake weighted projective spaces arising from the Batyrev-Borisov construction, which allows us in particular to determine explicitly all those of dimension up to five. Our approach in based on a new classification algorithm for reflexive simplices, which we present as well. As a byproduct of our methods, we obtain explicit formulae for the Picard group and the Gorenstein index of any fake weighted projective space.