Universal Code Families

Let $E$ be a finite alphabet of $q$ elements and let $U_i$ be a subset of $E^n$, i.e., a $q$-ary code of length $n$ with a certain minimum Hamming distance $d_i=d(U_i)$. We call a family of such codes $U_1,\dots, U_s$ of length $n$ with distances $d_1,\dots,d_s$ universal if for any $i, j\in\{1,\dots,s\}$, $i\neq j$, and any code vectors $u\in U_i$, $u'\in U_j$, the distance $d(u, u')$ between them satisfies the condition
$$ d(u, u')\geq(d_i+d_j)/2. $$
We construct asymptotically optimal universal code families.