List-decodable zero-rate codes

TL;DR

This paper investigates the asymptotic behavior of list-decodable zero-rate codes in binary and Euclidean spaces, establishing convergence rates and extending classical bounds for various list sizes.

Contribution

It provides new bounds on the convergence rate of maximal decoding radius for list-decodable codes, extending classical results to larger list sizes and spherical codes.

Findings

Binary case: rate is Θ(M^{-1}) for even L

Binary case: rate is Θ(M^{-2/3}) for L=3

Spherical codes: rate is between Ω(M^{-1}) and O(M^{-2L/(L^2-L+2)})

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 work, 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 $\Theta(M^{-\tfrac{2}{3}})$. For the similar question about spherical codes, we prove the rate is $\Omega(M^{-1})$ and $O(M^{-\tfrac{2L}{L^2-L+2}})$.