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This paper presents the central finite-dimensional Hinfin filters for linear systems with state delay, that are suboptimal for a given threshold gamma with respect to a modified Bolza-Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the results previously obtained for linear time delay systems, the paper reduces the original Hinfin filtering problems to H2 (optimal mean-square) filtering problems, using the technique proposed in . The paper first presents the central suboptimal Hinfin filter for linear systems with state delay, based on the optimal H2 filter from , which contains a finite number of the filtering equations for any fixed filtering horizon, but this number grows unboundedly as time goes to infinity. To overcome that difficulty, the alternative central suboptimal Hinfin filter is designed for linear systems with state delay, which is based on the alternative optimal H2 filter from . Numerical simulations are conducted to verify performance of the designed central suboptimal filters for linear systems with state delay against the central suboptimal Hinfin filter available for linear systems without delays.