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The Bragg reflection waveguide which have up to date been investigated are usually formed by two identical or different Bragg reflectors which are separated by a low index guiding layer. Some significant effects have been obtained in the BRW with periodic mirrors, particularly: i) the each guided mode has two cutoff points as a result waveguide can be designed to support only the higher-order modes; ii) the possibility of losing a specific mode due to shrinking of the forbidden bandgap into point; iii) the appearance of modes with minus mode order m with using a thin guiding layer; iv) the high dispersion in such waveguides generates slow light in the vicinity of the PBG edges. At the same time the nonperiodic design of layered reflectors, quasiperiodic or fractal arrangements of their constituents, gives an additional degree of freedom which allows achieving an enhancement of the optical performance in some specific applications. In the present paper we propose symmetric aiperiodic Bragg reflection waveguide which consist of a low-index guiding layer sandwiched between two aperiodic layered mirrors. Here we assumed that both mirrors made up of stacks of alternating layers which are arranged according Thue-Morse or Kolakoski substitutional rules. The transmission characteristics of the guided modes of the proposed quasiperiodic BRW are detailed studied in this paper. In additional influence of changing of the waveguide parameters on the dispersion characteristics of the guided modes is presented. We suppose that this approach would afford much more flexibility for designing and varying BRW structures.