A Lower Bound on List Size for List Decoding

A
q
-ary error-correcting code
C
⊆ {1,2,...,
q
}
n
is said to be list decodable to radius ρ with list size
L
if every Hamming ball of radius ρ contains at most
L
codewords of
C
. We prove that in order for a
q
-ary code to be list-decodable up to radius (1-1/
q
)(1- ε)
n
, we must have
L
= Ω(1/ ε
²
) . Specifically, we prove that there exists a constant
cq
> 0 and a function
fq
such that for small enough ε > 0, if
C
is list-decodable to radius (1-1/
q
)(1- ε)
n
with list size
cq
/ ε
²
, then
C
has at most
fq
( ε) codewords, independent of
n
. This result is asymptotically tight (treating
q
as a constant), since such codes with an exponential (in
n
) number of codewords are known for list size
L
=
O
(1/ ε
²
). A result similar to ours is implicit in Blinovsky ( Problems of Information Transmission, 1986) for the binary (
q
=2) case. Our proof is simpler and works for all alphabet sizes, and provides more intuition for why the lower bound arises.