Threshold Rates of Code Ensembles: Linear Is Best

In this work, we prove new results concerning the combinatorial properties of random linear codes. By applying the thresholds framework from Mosheiff et al. (FOCS 2020) we derive fine-grained results concerning the list-decodability and -recoverability 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. This is analogous to a lower bound for list-decoding that was recently obtained by Guruswami et al. (IEEE TIT 2021B). As a comparison, we also pin down the list size of random codes which is
$\frac {\log _{q}\binom {q}{\ell }}{\varepsilon }$
. This result almost closes the
$O\left({\frac {q\log L}{L}}\right)$
gap left by Guruswami et al. (IEEE TIT 2021A). 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: 1) List-of-3 decodability of random linear codes over
$\mathbb {F}_{2}$
; 2) 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. A conclusion that we draw from our work is that, for many combinatorial properties of interest, random linear codes actually perform better than uniformly random codes, in contrast to the apparently standard intuition that uniformly random codes are best.