A catalog of complexity classes
Handbook of theoretical computer science (vol. A)
Approximation algorithms for NP-hard problems
The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Algebraic-Geometric Codes
The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Two theorems on random polynomial time
SFCS '78 Proceedings of the 19th Annual Symposium on Foundations of Computer Science
The intractability of computing the minimum distance of a code
IEEE Transactions on Information Theory
A new PCP outer verifier with applications to homogeneous linear equations and max-bisection
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
The inapproximability of lattice and coding problems with preprocessing
Journal of Computer and System Sciences - Special issue on computational complexity 2002
ACM Transactions on Algorithms (TALG)
Hardness of approximating the shortest vector problem in lattices
Journal of the ACM (JACM)
Designs, Codes and Cryptography
Algorithmic problems for metrics on permutation groups
SOFSEM'08 Proceedings of the 34th conference on Current trends in theory and practice of computer science
Subspace polynomials and limits to list decoding of Reed-Solomon codes
IEEE Transactions on Information Theory
A simple deterministic reduction for the gap minimum distance of code problem
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Better inapproximability results for maxclique, chromatic number and min-3lin-deletion
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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We show that the minimum distance of a linear code (or equivalently, the weight of the lightest code-word) is not approximable to within any constant factor in random polynomial time (RP), unless NP equals RP. Under the stronger assumption that NP is not contained in RQP (random quasi-polynomial time), we show that the minimum distance is not approximable to within the factor \math, for any \math, where n denotes the block length of the code.Our results hold for codes over every finite field, including the special case of binary codes. In the process we show that the nearest code-word problem is hard to solve even under the promise that the number of errors is (a constant factor) smaller than the distance of the code. This is a particularly meaningful version of the nearest code-word problem.Our results strengthen (though using stronger assumptions) a previous result of Vardy who showed that the minimum distance is NP-hard to compute exactly. 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.