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.