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.