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We give a branch-and-cut algorithm for finding the minimum distance of a binary linear block code. We give two integer programming (IP) models and study the convex hull of the single constraint relaxation of these IP models. We use the new inequalities as cuts in a branch-and-cut scheme. Finally, we report computational results based on turbo and low density parity check (LDPC) codes that demonstrate the effectiveness of our cuts. We demonstrate that our IP formulation and specific cuts are efficient tools for determining the minimum distance of moderate size linear block codes, specifically, they are very efficient for LDPC codes, and provide us with an additional tool for solving this important problem.