Maximum Distance Separable Codes Based on Circulant Cauchy Matrices

We present a maximum-separable-distance (MDS) code suitable for computing erasure resilient codes for large word lengths. Given n data blocks (words) of any even bit length w the Circulant Cauchy Codes compute m ≤ w+1 code blocks of bit length w using XOR-operations, such that every combination of n data words and code words can reconstruct all data words. The number of XOR bit operations is at most 3nmw for encoding all check blocks. The main contribution is the small bit complexity for the reconstruction of u ≤ m missing data blocks with at most 9nuw XOR operations. We show the correctness for word lengths of form w = p − 1 where p is a prime number for which two is a primitive root. We call such primes Artin numbers. We use efficiently invertible Cauchy matrices in a finite field GF [2] for computing the code blocks To generalize these codes for all even word lengths w we use independent encodings by partitioning each block into sub-blocks of size pi − 1, i.e. w = ∑ i=1 pi − for Artin numbers pi. While it is not known whether infinitely many Artin numbers exist we enumerate all Circulant Cauchy Codes for w ≤ 10 yielding MDS codes for all m+ n ≤ 10 62 w.