Hardness of Approximating the Minimum Distance of a Linear Code

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 2log ( ) , 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 assumption...

• Type: Conference
• Year: 1999
• Where: FOCS
• Authors: Ilya Dumer, Daniele Micciancio, Madhu Sudan
• Keywords: FOCS 1999, Minimum Distance, Polynomial Time, Stronger Assumptions, Theoretical Computer Science