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Hardness of approximating the minimum distance of a linear code

By Ilya Dumer, Daniele Micciancio and Madhu Sudan


Abstract—We show that the minimum distance of a linear code is not approximable to within any constant factor in random polynomial time (RP), unless nondeterministic polynomial time (NP) equals RP. We also show that the minimum distance is not approximable to within an additive error that is linear in the block length of the code. Under the stronger assumption that NP is not contained in random quasi-polynomial time (RQP), we show that the minimum distance is not approximable to within the factor P �� � @ A, for any H. Our results hold for codes over any finite field, including binary codes. In the process, we show that it is hard to find approximately nearest codewords even if the number of errors exceeds the unique decoding radius P by only an arbitrarily small fraction. We also prove the hardness of the nearest codeword problem for asymptotically good codes, provided the number of errors exceeds @P QA. Our results for the minimum distance problem strengthen (though using stronger assumptions) a previous result of Vardy who showed that the minimum distance cannot be computed exactly in deterministic polynomial time (P), unless P a NP. Our results are obtained by adapting proofs of analogous results for integer lattices due to Ajtai and Micciancio. A critical component in the adaptation is our use of linear codes that perform better than random (linear) codes. Index Terms—Approximation algorithms, bounded distance decoding, computational complexity, dense codes, linear codes, minimum-distance problem, NP-hardness, relatively near codeword problem. I

Year: 1999
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