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GNSS is now being speedily expanded to our daily life, but the positioning precision still can hardly meet the demands of many high-precision applications, such as approaching landing system on airports. Due to the development of GNSS, quadruple-frequency signals are now available in China's BeiDou Navigation Satellite System (BDS) and the European Galileo system, which can contribute to positioning precision. Positioning precision can not be improved obviously by quadruple-frequency carrier phases until cycle slips are detected and repaired. A method using four linear combinations to detect and repair quadruplefrequency cycle slips is proposed in the paper. The choices of the four linear combinations are conducted in cascaded steps in accordance to the cycle slip fixing probability. When the four detection combinations are determined, cycle slips on original carrier phase observations can be uniquely determined. The proposed algorithm has been tested on real 30-second quadruple-frequency static observations of BDS and Galileo and on real 0.05-second quadruple-frequency kinematic observations of BDS and Galileo. Simulated and real cycle slips are tested. The results show that the proposed algorithm can detect and repair cycle slips even for one cycle effectively.INDEX TERMS Quadruple-frequency observations, cycle slip, linear combination, BDS, Galileo.

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