I think what you are looking for is mixed codes.
A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k=2$ while $n_{k+1}=\cdots=n_N=3$. Brouwer keep an online list of known 3/2 mixed code.
I think they also talked about some general cases. Anyway, more papers can be found from the references therein or by the key word. I also find this online list with a 4/3/2 covering code.
There is a mysterious paper titled "Mixed Codes: Bounds, Constructions and Some Applications" by Perkins--Sakhnovich--Smith, appearently "submitted for publication", but I can not find it.