Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery

Abstract

In this work we consider the list-decodability and list-recoverability of arbitrary q-ary codes, for all integer values of $q\geq 2$ . A code is called $(p,L)_{q}$ -list-decodable if every radius pn Hamming ball contains less than L codewords; $(p,\ell ,L)_{q}$ -list-recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length $\ell $ 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 $(p,\ell ,L)_{q}$ -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_{*}+\varepsilon $ fraction of errors must have size $O_{\varepsilon }(1)$ , 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_{*}-\varepsilon $ 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 demonstrating 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.