An Encoding Algorithm of Triply Extended Reed–Solomon Codes With Asymptotically Optimal Complexities

In this paper, we devise a fast encoding algorithm for triply extended Reed-Solomon codes. The proposed approach requires approximately two XORs per bit, which improves the prior result of three XORs per bit established by certain maximum distance separable (MDS) array codes. We also prove that, for MDS codes with two and three parities, the scheduling algorithms require at least two XORs per bit. To the best of our knowledge, this is the first provable scheduling algorithm for the triple-parity MDS codes to approach the theoretical lower bounds. The implementation with SIMD instructions is provided. The simulations show that the proposed approach is competitive, as compared with other cutting edge implementations.