  In this work, we prove new results concerning the combinatorial properties of
random linear codes.
  Firstly, we prove a lower bound on the list-size required for random linear
codes over $\mathbb F_q$ $\varepsilon$-close to capacity to list-recover with
error radius $\rho$ and input lists of size $\ell$. We show that the list-size
$L$ must be at least $\frac{\log_q\binom{q}{\ell}-R}{\varepsilon}$, where $R$
is the rate of the random linear code. As a comparison, we also pin down the
list size of random codes which is $\frac{\log_q\binom{q}{\ell}}{\varepsilon}$.
This leaves open the possibility (that we consider likely) that random linear
codes perform better than random codes for list-recoverability, which is in
contrast to a recent gap shown for the case of list-recovery from erasures
(Guruswami et al., IEEE TIT 2021B).
  Next, we consider list-decoding with constant list-sizes. Specifically, we
obtain new lower bounds on the rate required for list-of-$3$ decodability of
random linear codes over $\mathbb F_2$; and list-of-$2$ decodability of random
linear codes over $\mathbb F_q$ (for any $q$). This expands upon Guruswami et
al. (IEEE TIT 2021A) which only studied list-of-$2$ decodability of random
linear codes over $\mathbb F_2$. Further, in both cases we are able to show
that the rate is larger than that which is possible for uniformly random codes.
