Abstract—The list-decodability of random linear codes is shown to be as good as that of general random codes. Specifically, for every fixed finite field Fq, p ∈ (0, 1 − 1/q) and ε>0, itis proved 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 listsize of O(1/ε) suffices to have rate within ε of the informationtheoretically optimal rate of 1−Hq(p). The best previously known list-size bound was q O(1/ε) (except in the q =2case where a listsize bound of O(1/ε) was known). The main technical ingredient in the proof is a strong upper bound on the probability that ℓ random vectors chosen from a Hamming ball centered at the origin have too many (more than Ω(ℓ)) vectors from their linear span also belong to the ball
To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.