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We propose a new implementation of the block Lanczos–Montgomery method with reduced data exchanges for the solution of large linear systems over finite fields. The theoretical estimates obtained for parallel complexity indicate that the data exchanges in the proposed implementation for record-high matrix sizes and block sizes 50 require less time than those in ideally parallelizable computations with dense blocks. According to numerical results, the acceleration depends almost linearly on the number of cores (up to 2,000 cores). Then, the dependence becomes close to the square root of the number of cores.