  We present some blocks of Ariki-Koike algebras $\mathcal{H}_{n,r}$ for which
the decomposition matrices are independent of the characteristic of the
underlying field. We complete the description of the graded decomposition
numbers for blocks of Ariki-Koike algebras of weight at most two, which
consists of analysing the indecomposable core blocks at level $r = 3$, and give
a closed formula for the decomposition numbers in this case.
