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The coefficient decimation technique for reconfigurable FIR filters was recently proposed as a filter structure with low computational complexity. In this brief, we propose to design these filters using linear programming taking all configuration modes into account, instead of only considering the initial reconfiguration mode as in previous works. Minimax solutions with significantly lower approximation errors compared to the straightforward design method in earlier works are obtained. In addition, some new insights that are useful when designing coefficient decimation filters are provided.