13 research outputs found
A Lower Bound on List Size for List Decoding
A q-ary error-correcting code C ā {1, 2,..., q} n is said to be list decodable to radius Ļ with list size L if every Hamming ball of radius Ļ 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 ā 1/q)(1 ā Īµ)n, we must have L = ā¦(1/Īµ 2). Specifically, we prove that there exists a constant cq> 0 and a function fq such that for small enough Īµ> 0, if C is list-decodable to radius (1 ā 1/q)(1 ā Īµ)n with list size cq/Īµ 2, then C has at most fq(Īµ) 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/Īµ 2). A result similar to ours is implicit in Blinovsky [Bli1] for the binary (q = 2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.