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...

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Veröffentlicht in:IEEE transactions on information theory Jg. 58; H. 11; S. 6935 - 6941
Hauptverfasser: Cheng, Qi, Wan, Daqing
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York, NY IEEE 01.11.2012
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Applied sciences
Approximation
Approximation algorithm
Approximation algorithms
Coding
coding theory
Coding, codes
Error correcting codes
Exact sciences and technology
Hamming codes
Information theory
Information, signal and communications theory
Lattices
Linear code
minimum distance problem
NP-complete
Polynomials
Projection
Reduction
Reed-Solomon codes
Signal and communications theory
Subspaces
Telecommunications and information theory
Vectors
Reed Solomon code
Hamming distance
Subspace method
Carrier to noise ratio
NP-complete
Polynomial method
Approximation algorithm
Polynomial time
Linear code
coding theory
Coding
NP hard problem
NP complete problem
Minimal distance
Deterministic approach
minimum distance problem
Hadamard codes
Deterministic algorithms
ISSN:0018-9448, 1557-9654
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Zusammenfassung: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.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2012.2209198