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