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This paper proposes two improved triple parity placement schemes, the HDD1 (horizontal and dual diagonal) and HDD2 schemes, to enhance the reliability of a RAID system. Both schemes can tolerate up to three disk failures by using three types of parity information (horizontal, diagonal, and anti-diagonal parities) in RAID disk block partitions. The HDD1 scheme can reduce the occurrences of bottlenecks because its horizontal and anti-diagonal parities are uniformly distributed over a disk array, while diagonal parities are placed in a dedicated disk. The HDD2 scheme uses one more disk than HDD1 to store the horizontal parities and an additional diagonal parity, while the anti-diagonal and the diagonal parities are placed in the same way as in the HDD1 scheme, only with a minor difference. The encoding and decoding algorithms of both schemes are simple and effective. Many of the steps of the encoding and decoding algorithms can be executed in parallel. Both schemes enable a RAID to recover rapidly from up to three disk failures, with a single algorithm applied straightforwardly.