Graded decomposition numbers of Ariki-Koike algebras for blocks of small weight

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.