Block error-correcting codes over inhomogeneous alphabets For $n := (n_1,\dots,n_N) \in \mathbb{N}_{>1}^N$, let $X_n := \prod_{j=1}^N [n_j]$, where as usual $[m] := \{1,\dots,m\}$. 

Are there any known generic constructions for (Hamming) sphere packings in $X_n$ other than the "trivial" ones that essentially embed each factor $[n_j]$ in some $\mathbb{F}_{p^{r(j)}}$ for $p$ fixed and use a $p$-ary block error-correcting code?

Note that if $n_j = q$ for all $j$ then the problem becomes one of "merely" finding good $q$-ary block error-correcting codes. The "trivial" construction above shows that the general problem posed above can also be embedded in this classical problem, albeit inelegantly and probably far from optimally.
 A: As per Hao Chen's comments, check out Lenstra's work on number field codes. Guruswami did some follow-up work couple decades later.
A: 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.
Another interesting paper is Perkins--Sakhonivich--Smith.  It seems to be initially cited as "mixed codes: bounds, constructions and some applications" before publication, which confused me. Fujiwara also find this reference.
Anyway, more papers can be found from the references therein or by the key word.  I also find this online list with 4/3/2 mixed covering codes and many references.
update: Turbo mentioned a work of Lenstra in the comment.  It already uses the term "mixed codes" on the first page.
A: As mentioned in Hao Chen's answer, what you're looking for seems to be a good mixed code. There don't seem to be many papers on this. But apparently the following paper gives the best known general upper bound on the code side:
S. Perkins, A. L. Sakhnovich, D. H. Smith, On an upper bound for mixed error-correcting codes, IEEE Transactions on Information Theory, 52 (2006), 708--712
The results given there are a little cumbersome to spell out, and I'm not an expert on this at all. So, please check what's in there for yourself. Perhaps, this is (part of) the ``mysterious paper'' Hao Chen is talking about.
