Limits to List Decoding Random Codes

Abstract

It has been known since [Zyablov and Pinsker 1982] that a random q-ary code of rate 1 − H _q (ρ) − ε (where 0 < ρ< 1 − 1/q, ε> 0 and H _q (·) is the q-ary entropy function) with high probability is a (ρ,1/ε)-list decodable code. (That is, every Hamming ball of radius at most ρn has at most 1/ε codewords in it.) In this paper we prove the “converse” result. In particular, we prove that for every 0 < ρ< 1 − 1/q, a random code of rate 1 − H _q (ρ) − ε, with high probability, is not a (ρ,L)-list decodable code for any, where c is a constant that depends only on ρ and q. We also prove a similar lower bound for random linear codes.