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.



Block error-correcting codes over inhomogeneous alphabets
I think what you are looking for is **mixed codes**.


A good start point would be [Brouwer--Hämäläinen--Östergård--Sloane](http://www.ams.org/mathscinet-getitem?mr=1486654). 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](http://www.win.tue.nl/~aeb/codes/23codes.html). I think they also talked about some general cases.


Another interesting paper is [Perkins--Sakhonivich--Smith](http://www.ams.org/mathscinet-getitem?mr=2236184). 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](http://www.sztaki.hu/~keri/codes/) with 4/3/2 mixed covering codes and many references.


**update**: Turbo mentioned a work of [Lenstra](http://www.ams.org/mathscinet-getitem?mr=878574) in the comment. It already uses the term "mixed codes" on the first page.


