A Lower Bound on List Size for List Decoding

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–e)n, we must have L = Ω(1/e2). Specifically, we prove that there exists a constant cq>0 and a function fq such that for small enough e > 0, if C is list-decodable to radius (1–1/q)(1–e)n with list size cq/e2, then C has at most fq(e) 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/e2). 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.