Erasure Codes for Reading and Writing (Technical Report tr-ri-06-274)

We introduce the Read-Write-Coding-System (RWC) – a very flexible class of MDS codes using a variation of Reed-Solomon (RS) codes, but offer enhanced possibilities to update any codeword, and to adjust the redundancy and thus, the fault-tolerance capability of the code. In particular, given a message x of length k, a block code y of length n and arbitrarily chosen parameters k ≤ r, w ≤ n, our RW codes provide a linear (n, k, d)code such that (a) the minimum distance d = n − r + 1 for any two codewords, and (b) any two codewords are within a distance of at most w from each other. The parameter r ≥ k determines the rate of the code and thus, the minimum fraction of encoded blocks needed to recover x, and w ≤ n determines the maximum number of blocks to update y if x is modified. This is different from RS codes which always need to rewrite y completely if x changes. In addition, for a subset of the RW codes, all mentioned parameters are adjustable even online such that the code becomes adaptive to changing demands. RW codes are very attractive for e.g. storage networks in which, on one hand, I/O accesses are very costly, and on the other hand, redundancy is crucial. With an RW code, this trade-off can then be better adjusted than with RS codes. We state a tight lower bound for all RWC and show under which conditions of the symbol alphabet size RW codes do exist. Furthermore, we point out some useful properties regarding safety and security of the data encoded.