Subadditivity in t for all-symbol PIR and batch codes

Let $ASP(k,t,q)$ and $ASB(k,t,q)$ denote the minimum lengths of, respectively, all-symbol PIR and all-symbol batch codes with parameters $k,t,q$. For all $k,q$ and $t_1,t_2\geq 1$, Subadditivity in $t$.

$$ASP(k,t_1+t_2,q)\leq ASP(k,t_1,q)+ASP(k,t_2,q),$$ $$ASB(k,t_1+t_2,q)\leq ASB(k,t_1,q)+ASB(k,t_2,q).$$

The claim extends the known subadditivity results for functional PIR and batch codes; the paper notes that the corresponding general statement for all-symbol parameters remains open.