List-decodable zero-rate codes

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 τ ϵ [0,1] for which there exists an arrangement of M balls of relative Hamming radius τ 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 → ∞ the maximal τ decreases to a well-known critical value τ _L . In this paper, we prove several results on the rate of this convergence. For the binary case, we show that the rate is Θ (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 Θ (M ^-(2/3) ). For the similar question about spherical codes, we prove the rate is Ω (M ^-1 ) and O(M- ^{(2L/L 2 -L+2)} ).