We show that the list-decodability of random linear codes is as good as that of general random codes. Specifically, for every fixed finite field Fq, p ∈ (0, 1 − 1/q) and ε > 0, we prove that with high probability a random linear code C in F n q of rate (1 − Hq(p) − ε) can be list decoded from a fraction p of errors with lists of size at most O(1/ε). This also answers a basic open question concerning the existence of highly list-decodable linear codes, showing that a list-size of O(1/ε) suffices to have rate within ε of the "list decoding capacity" 1 − Hq(p). The best previously known list-size bound was q O(1/ε) (except in the q = 2 case where a list-size bound of O(1/ε) was known). The main technical ingredient in our proof is a strong upper bound on the probability that random vectors chosen from a Hamming ball centered at the origin have too many (more than Θ( )) vectors from their linear span also belong to the ball.
When ρ is constant, Guruswami, Håstad, and Kopparty REF show that a random linear code of rate 1−Hq
On the list-decodability of random linear codes
STOC '10
Mathematics Computer Science