A deterministic reduction for the gap minimum distance problem

Abstract

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 in [14]. In [8], the gap version of the problem was shown to be NP-hard for any constant factor under a randomized reduction. It was shown in the same paper that the minimum distance problem is not approximable in randomized polynomial time to the factor 2log1-e 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 2log1-en unless NP ⊆ DTIME(2polylog(n)). As the main technical contribution, for any constant 2/3