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Digital fingerprinting, a copyright protection technique for digital contents, is considered. Digital fingerprinting should deter collusion attacks, where several fingerprinted copies of the same content are mixed to disturb their fingerprints. In this paper, we consider the averaging attack, which has effect for multimedia fingerprinting. We propose new collusion-secure fingerprinting codes based on finite geometries (FGs) which increase the rate of conventional collusion-secure codes, while they guarantee to identify the same number of colluders. Due to the new FG-based fingerprinting codes, the system can deal with a larger number of users to distribute a digital content.