List-Decodable Zero-Rate Codes

Abstract

We consider list decoding in the zero-rate regime for two cases—the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal $\tau \in [{0,1}]$ for which there exists an arrangement of $M$ balls of relative Hamming radius $\tau $ in the binary hypercube (of arbitrary dimension) with the property that no point of the latter is covered by $L$ or more of them. As $M\to \infty $ the maximal $\tau $ decreases to a well-known critical value $\tau _{L}$ . In this paper, we prove several results on the rate of this convergence. For the binary case, we show that the rate is $\Theta (M^{-1})$ when $L$ is even, thus extending the classical results of Plotkin and Levenshtein for $L=2$ . For $L=3$ , the rate is shown to be $\Theta (M^{-({2}/{3})})$ . For the similar question about spherical codes, we prove the rate is $\Omega (M^{-1})$ and $O(M^{-({2L}/{L^{2}-L+2})})$ .