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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| Vydáno v: | IEEE transactions on information theory Ročník 58; číslo 11; s. 6935 - 6941 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York, NY
IEEE
01.11.2012
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| 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 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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| AbstractList | 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 log 1 - epsilon n unless NP [subE] RTIME ( 2 polylog ( 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 log 1 - epsilon n unless NP [subE] DTIME ( 2 polylog ( n ) ) . As the main technical contribution, for any constant 2 / 3 < rho < 1 , we present a deterministic algorithm that given a positive integer s , runs in time poly ( s ) and constructs a code cal C of length poly ( s ) with an explicit Hamming ball of radius rho d ( cal 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 ( cal C ) denotes the minimum distance of cal C . The codes are obtained by concatenating Reed-Solomon codes with Hadamard codes. 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 $2^{log^{1-epsilon}n}$ unless $NPsubseteq RTIME(2^{polylog(n)})$. 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 $2^{log^{1-epsilon}n}$ unless $NPsubseteq DTIME(2^{polylog(n)})$. As the main technical contribution, for any constant $2/3 < rho < 1$, we present a deterministic algorithm that given a positive integer $s$ , runs in time $poly(s)$ and constructs a code ${cal C}$ of length $poly(s)$ with an explicit Hamming ball of radius $rho d({cal C})$, such that the projection at the first $s$- /formula> coordinates sends the codewords in the ball surjectively onto a linear subspace of dimension $s$ , where $d({cal C})$ denotes the minimum distance of ${cal C}$. The codes are obtained by concatenating Reed-Solomon codes with Hadamard codes. [PUBLICATION 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 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. |
| Author | Qi Cheng Daqing Wan |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | A Deterministic Reduction for the Gap Minimum Distance Problem |
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