  We analyze the list-decodability, and related notions, of random linear
codes. This has been studied extensively before: there are many different
parameter regimes and many different variants. Previous works have used
complementary styles of arguments---which each work in their own parameter
regimes but not in others---and moreover have left some gaps in our
understanding of the list-decodability of random linear codes. In particular,
none of these arguments work well for list-recovery, a generalization of
list-decoding that has been useful in a variety of settings.
  In this work, we present a new approach, which works across parameter regimes
and further generalizes to list-recovery. Our main theorem can establish better
list-decoding and list-recovery results for low-rate random linear codes over
large fields; list-recovery of high-rate random linear codes; and it can
recover the rate bounds of Guruswami, Hastad, and Kopparty for constant-rate
random linear codes (although with large list sizes).
