  We study the list-decodability and list-recoverability of two code ensembles:
random linear codes and random Reed-Solomon codes. Inspired by the existing
research about local properties of random code ensembles over small alphabets,
we develop a new framework to study a similar family of properties over larger
alphabets, such as in the case of Reed-Solomon codes. We introduce the notion
of local coordinate-wise linear (LCL) properties, which encompasses various
natural properties including list-decodability and list-recoverability.
  Our main contributions are threefold: (1) we establish a threshold theorem
for LCL properties of random linear codes, identifying a critical rate below
which these codes almost surely satisfy a given property and above which they
almost surely do not; (2) we use this framework to prove that random linear
codes are essentially optimally list-decodable and list-recoverable in the
large alphabet regime; and (3) we demonstrate a reduction from random linear
codes to random Reed-Solomon codes, showing that Reed-Solomon codes inherit the
LCL properties of linear codes with similar rates. We conclude that random
Reed-Solomon codes, with alphabet size that may be as small as linear in the
code length, are also optimally list-decodable and list-recoverable.
  For list-recovery, our results about both ensembles constitute a significant
leap from the best previously known bounds. For list-decodability, we recover
recent theorems via a more direct approach, avoiding reliance on complex
external results.
