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### Paper:

TR19-159 | 11th November 2019 19:06

#### SETH-hardness of Coding Problems

TR19-159
Authors: Noah Stephens-Davidowitz, Vinod Vaikuntanathan
Publication: 11th November 2019 21:23
We show that assuming the strong exponential-time hypothesis (SETH), there are no non-trivial algorithms for the nearest codeword problem (NCP), the minimum distance problem (MDP), or the nearest codeword problem with preprocessing (NCPP) on linear codes over any finite field. More precisely, we show that there are no NCP, MDP, or NCPP algorithms running in time $q^{(1-\epsilon)n}$ for any constant $\epsilon>0$ for codes with $q^n$ codewords. (In the case of NCPP, we assume non-uniform SETH.)
We also show that there are no sub-exponential-time algorithms for $\gamma$-approximate versions of these problems for some constant $\gamma > 1$, under different versions of the exponential-time hypothesis.