A Lower Bound on List Size for List Decoding

Abstract

A q-ary error-correcting code C ⊆ {1,2,...,q}ⁿ is said to be list decodable to radius ρ with list size L if every Hamming ball of radius ρ contains at most L codewords of C. We prove that in order for a q-ary code to be list-decodable up to radius (1–1/q)(1–ε)n, we must have L = Ω(1/ε ²). Specifically, we prove that there exists a constant c _q >0 and a function f _q such that for small enough ε > 0, if C is list-decodable to radius (1–1/q)(1–ε)n with list size c _q /ε ², then C has at most f _q (ε) codewords, independent of n. This result is asymptotically tight (treating q as a constant), since such codes with an exponential (in n) number of codewords are known for list size L = O(1/ε ²).

A result similar to ours is implicit in Blinovsky [Bli] for the binary (q=2) case. Our proof works for all alphabet sizes, and is technically and conceptually simpler.