The passage discusses the NP-completeness of certain coding problems, specifically the decoding problem for linear codes and the problem of finding the weights of a linear code. It shows that both problems are NP-complete, suggesting that no polynomial-time algorithm exists for solving them. The authors provide a detailed proof by reducing these problems to the NP-complete problem of three-dimensional matching. They also explore related problems, such as the SUBSPACE WEIGHTS and COSET WEIGHTS problems, and show that they are also NP-complete. The paper concludes with remarks on the implications of these findings and suggests further research directions, particularly on the problem of finding the minimum weight of a general code.The passage discusses the NP-completeness of certain coding problems, specifically the decoding problem for linear codes and the problem of finding the weights of a linear code. It shows that both problems are NP-complete, suggesting that no polynomial-time algorithm exists for solving them. The authors provide a detailed proof by reducing these problems to the NP-complete problem of three-dimensional matching. They also explore related problems, such as the SUBSPACE WEIGHTS and COSET WEIGHTS problems, and show that they are also NP-complete. The paper concludes with remarks on the implications of these findings and suggests further research directions, particularly on the problem of finding the minimum weight of a general code.