  We consider the dimensions of a family of self-affine sets related to the
Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and
then randomise the translation vectors with the stipulation that the column
structure is preserved. As such, we maintain one of the key features in the
Bedford-McMullen set up in that alignment causes the dimensions to drop from
the affinity dimension. We compute the Hausdorff, packing and box dimensions
outside of a small set of exceptional translations, and also for some explicit
translations even in the presence of overlapping. Our results rely on, and can
be seen as a partial extension of, M. Hochman's recent work on the dimensions
of self-similar sets and measures.
