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

TL;DR

This paper precisely determines the maximum error fraction (zero-rate threshold) for list-recoverable codes over any alphabet size, establishing bounds that delineate when codes must be small or can have positive rate.

Contribution

It introduces the exact zero-rate threshold for list-recoverability, proving a Plotkin bound for these codes, and corrects a flaw in previous related proofs.

Findings

Zero-rate threshold $p_*$ computed for all $q$-ary codes.

Codes correcting errors beyond $p_*$ must be of bounded size.

Existence of positive rate codes below $p_*$ via random constructions.

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.