Limits to List Decoding Random Codes

It has been known since [Zyablov and Pinsker 1982] that a random q-ary code of rate 1 − Hq(ρ) − ε (where 0 < ρ < 1 − 1/q, ε > 0 and Hq(·) 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−Hq(ρ)−ε, with high probability, is not a (ρ, L)-list decodable code for any L 6 c ε , where c is a constant that depends only on ρ and q. We also prove a similar lower bound for random linear codes. Previously, such a tight lower bound on the list size was only known for the case when ρ > 1− 1/q−O(√ε) [Guruswami and Vadhan, 2005]. For binary codes a lower bound is known for all 0 < ρ < 1/2, though the lower bound is asymptotically weaker than our bound [Blinovsky, 1986]. These results however are not subsumed by ours as they hold for arbitrary codes of rate 1−Hq(ρ)− ε. ∗Research supported in part by startup funds from University at Buffalo. Electronic Colloquium on Computational Complexity, Report No. 13 (2009)