Unlike conventional disk array systems such as redundant arrays of independent disks (RAID), distributed storage systems do not require a central controller for a group of disk drives. This decentralized structure avoids a single point of failure as well as bottlenecks in data transfers. The structure should therefore be suitable for dependable, large-scale storage systems. This paper proposes new distributed erasure correction algorithms suitable for the distributed storage systems, and also proposes new classes of triple-erasure correcting codes designed using combinatorial theory such as the Steiner triple system. We briefly evaluate the proposed coding method in terms of the number of check disks, the number of disk accesses, and the mean time to data loss (MTTDL). The evaluation demonstrates that the proposed codes require fewer check disks than those in disk mirroring. The proposed erasure correction algorithms also require fewer disk accesses than erasure correction using Reed-Solomon codes. For a large storage system with 400 information disks, the proposed coding method with 37 check disks increases the MTTDL from 0.10 years to 167 years.