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<sup>log1-ϵ</sup><i>n</i> unless <i>NP</i> ⊆ <i>RTIME</i><sup>(2polylog(n)</sup>). 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 <i>P</i>=<i>NP</i>. We also prove that the minimum distance is not approximable in deterministic polynomial time to the factor 2<sup>log1-ϵ</sup><i>n</i> unless <i>NP</i> ⊆ <i>DTIME</i><sup>(2polylog(n)</sup>). As the main technical contribution, for any constant 2/3 <; ρ <; 1, we present a deterministic algorithm that given a positive integer <i>s</i> , runs in time <i>poly</i>(<i>s</i>) and constructs a code <i>C</i> of length <i>poly</i>(<i>s</i>) with an explicit Hamming ball of radius ρ<i>d</i>(<i>C</i>), such that the projection at the first <i>s</i> coordinates sends the codewords in the ball surjectively onto a linear subspace of dimension <i>s</i> , where <i>d</i>(<i>C</i>) denotes the minimum distance of <i>C</i>. The codes are obtained by concatenating Reed-Solomon codes with Hadamard codes.

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