Hardness of approximating the minimum distance of a linear code

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 2 log ( ) , for any 0. 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 2 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 (2 3) . 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 = 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.

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“…• The probabilistic construction of a similar gadget for linear codes used in [11] to prove the NP-hardness of MDP has been successfully derandomized [8]. This provides hope that a derandomization of the lattice gadget employed in this paper may be possible too.…”

“…Tensoring immediately yields inapproximability results for SVP within larger factors, still under randomized reductions with one-sided error. Moreover, our basic NP-hardness proof-within small approximation factors-is very similar to those in [20,11] so, as explained in the previous paragraphs, it may be more easily derandomized. We remark that the standard tensor product operation amplifies the approximation factor for any instance of the SVP variant defined in this paper, not just for the output of our basic reduction.…”

“…So, the techniques in [8,4] for the derandomization of the coding gadget of [11] may help to derandomize the construction of the lattice gadget described in this paper.…”

“…For example, the tensor product of linear codes is used in [11] to amplify the NP-hardness of approximating MDP to arbitrarily large constant factors. This suggests the use of tensor products of lattices to prove the NP-hardness of SVP within large constant factors, starting from the inapproximability result of [20] for factors below √ 2.…”

Tensor-based hardness of the shortest vector problem to within almost polynomial factors

“…For example, the tensor product of linear codes is used to amplify the NP-hardness of approximating the minimum distance in a linear code of block length n to arbitrarily large constants under polynomial-time reductions and to 2 (log n) 1−ε (for any ε > 0) under quasipolynomial-time reductions [16]. This example motivates one to use the tensor product of lattices to increase the hardness factor known for approximating SVP.…”