A Deterministic Reduction for the Gap Minimum Distance Problem

Determining the minimum distance of a linear code is one of the most important problems in algorithmic coding theory. The exact version of the problem was shown to be NP-complete by Vardy. The gap version of the problem was shown to be NP-hard for any constant factor under a randomized reduction in an earlier work. It was shown in the same paper that the minimum distance problem is not approximable in randomized polynomial time to the factor 2
^log1-ϵ
n
unless
NP
⊆
RTIME
^(2polylog(n)
). In this paper, we derandomize the reduction and thus prove that there is no deterministic polynomial time algorithm to approximate the minimum distance to any constant factor unless
P
=
NP
. We also prove that the minimum distance is not approximable in deterministic polynomial time to the factor 2
^log1-ϵ
n
unless
NP
⊆
DTIME
^(2polylog(n)
). As the main technical contribution, for any constant 2/3 <; ρ <; 1, we present a deterministic algorithm that given a positive integer
s
, runs in time
poly
(
s
) and constructs a code
C
of length
poly
(
s
) with an explicit Hamming ball of radius ρ
d
(
C
), such that the projection at the first
s
coordinates sends the codewords in the ball surjectively onto a linear subspace of dimension
s
, where
d
(
C
) denotes the minimum distance of
C
. The codes are obtained by concatenating Reed-Solomon codes with Hadamard codes.