Tight bounds on list-decodable and list-recoverable zero-rate codes

In this work, we consider the list-decodability and list-recoverability of codes in the zero-rate regime. Briefly, a code 𝒞 ⊆ [q]ⁿ is (p,𝓁,L)-list-recoverable if for all tuples of input lists (Y₁,… ,Y_n) with each Y_i ⊆ [q] and |Y_i| = 𝓁, the number of codewords c ∈ 𝒞 such that c_i ∉ Y_i for at most pn choices of i ∈ [n] is less than L; list-decoding is the special case of 𝓁 = 1. In recent work by Resch, Yuan and Zhang (ICALP 2023) the zero-rate threshold for list-recovery was determined for all parameters: that is, the work explicitly computes p_*: = p_*(q,𝓁,L) with the property that for all ε > 0 (a) there exist positive-rate (p_*-ε,𝓁,L)-list-recoverable codes, and (b) any (p_*+ε,𝓁,L)-list-recoverable code has rate 0. In fact, in the latter case the code has constant size, independent on n. However, the constant size in their work is quite large in 1/ε, at least |𝒞| ≥ (1/(ε))^O(q^L). Our contribution in this work is to show that for all choices of q,𝓁 and L with q ≥ 3, any (p_*+ε,𝓁,L)-list-recoverable code must have size O_{q,𝓁,L}(1/ε), and furthermore this upper bound is complemented by a matching lower bound Ω_{q,𝓁,L}(1/ε). This greatly generalizes work by Alon, Bukh and Polyanskiy (IEEE Trans. Inf. Theory 2018) which focused only on the case of binary alphabet (and thus necessarily only list-decoding). We remark that we can in fact recover the same result for q = 2 and even L, as obtained by Alon, Bukh and Polyanskiy: we thus strictly generalize their work. Our main technical contribution is to (a) properly define a linear programming relaxation of the list-recovery condition over large alphabets; and (b) to demonstrate that a certain function defined on a q-ary probability simplex is maximized by the uniform distribution. This represents the core challenge in generalizing to larger q (as a binary simplex can be naturally identified with a one-dimensional interval). We can subsequently re-utilize certain Schur convexity and convexity properties established for a related function by Resch, Yuan and Zhang along with ideas of Alon, Bukh and Polyanskiy.