This Letter presented a new-order optimal method for lowering the computational complexity of the traditional minimax design of digital finite impulse response (FIR) filter utilising linear programming (LP) in bandwidth interleaving digital-to-analogue converter (BI-DAC). In this order optimal method, a one-by-one increasing method was presented to optimise all of digital FIR subfilters' orders in BI-DAC so that these digital FIR subfilters' orders could be optimised to simultaneously satisfy all of given upper bounds of the distortion and aliasing errors. A design example was used to verify the effectiveness and low computational complexity of the presented low-complexity minimax design utilising LP, and the simulation results were well-pleasing.