Zero-rate thresholds and new capacity bounds for list-decoding and list-recovery

In this work we consider the list-decodability and list-recoverability of arbitrary q -ary codes, for all integer values of q ⩾ 2. A code is called ( p , L ) q -list-decodable if every radius pn Hamming ball contains less than L codewords; ( p , ℓ, L ) q -recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length ℓ and again stipulate that there be less than L codewords. Our main contribution is to precisely calculate the maximum value of p for which there exist infinite families of positive rate ( p , ℓ, L ) q -list-recoverable codes, the quantity we call the zero-rate threshold . Denoting this value by p *, we in fact show that codes correcting a p * + ε fraction of errors must have size O ε (1), i.e., independent of n . Such a result is typically referred to as a “Plotkin bound.” To complement this, a standard random code with expurgation construction shows that there exist positive rate codes correcting a p * − ε fraction of errors. We also follow a classical proof template (typically attributed to Elias and Bassalygo) to derive from the zero-rate threshold other tradeoffs between rate and decoding radius for list-decoding and list-recovery. Technically, proving the Plotkin bound boils down to demon-strating the Schur convexity of a certain function defined on the q -simplex as well as the convexity of a univariate function derived from it. We remark that an earlier argument claimed similar results for q -ary list-decoding; however, we point out that this earlier proof is flawed.