A New Construction and an Efficient Decoding Method for Rabin-Like Codes

Array codes have been widely used in communication and storage systems. To reduce computational complexity, one important property of the array codes is that only exclusive OR operations are used in the encoding and decoding processes. Cauchy Reed-Solomon codes, Rabin-like codes, and circulant Cauchy codes are existing Cauchy maximum-distance separable (MDS) array codes that employ Cauchy matrices over finite fields, circular permutation matrices, and circulant Cauchy matrices, respectively. All these codes can correct any number of failures; however, a critical drawback of existing codes is the high decoding complexity. In this paper, we propose a new construction of Rabin-like codes based on a quotient ring with a cyclic structure. The newly constructed Rabin-like codes have more supported parameters (prime p is extended to an odd number), such that the world sizes of them are more flexible than the existing Cauchy MDS array codes. An efficient decoding method using LU factorization of the Cauchy matrix can be applied to the newly constructed Rabin-like codes. It is shown that the decoding complexity of the proposed approach is less than that of existing Cauchy MDS array codes. Hence, the Rabin-like codes based on the new construction are attractive to distributed storage systems.