SETH-Hardness of Coding Problems

Abstract

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-ε)n for any constant ε>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 γ-approximate versions of these problems for some constant γ > 1, under different versions of the exponential-time hypothesis.