Efficient Repair of Reed-Solomon Codes

In this thesis we construct optimal/low repair bandwidth schemes for Reed-Solomon codes both in theoretical and practical scenarios. Our work on improving the repair bandwidth for repairing Reed-Solomon codes can help to speed up the recovery process in data distributed storage systems which is significant for enhancing data security, reducing data storage cost, or ensuring data transmission in the event of hardware failures. The first part of our research (Chapter 4) focuses on the optimal bandwidth repair schemes generated by algebraic constructions, including trace polynomials and subspace polynomials, to repair single and two erasures of full-length Reed-Solomon codes RS(n, k) over coding field $F_q^ℓ$. In terms of repairing single erasures, we recall the repair schemes constructed from trace polynomials by Guruswami and Wootters [5, 6], which are the very first proposed schemes for repairing Reed-Solomon codes following the trace repair approach, and providing a convincing illustration for the efficiency of trace repair in repairing Reed-Solomon codes. Next, we review the repair schemes generated from subspace polynomials to repair a failed node of Reed-Solomon codes proposed by Dau and Milenkovich [4, 7] showing the improvement on the range of the codes that can be repaired compared to the trace polynomial repair schemes. The focus then shifts to repair schemes for two erasures generated from subspace polynomials. In this part, we propose one-round repair schemes and multi-round repair schemes for two erasures with two phases (download phase and collaboration phase). The schemes with subspace polynomials offer a significant improvement in repair bandwidth compared to existing methods. More specifically, the subspace polynomial repair schemes can repair each failed node with at most $(n-1)(ℓ-m)$ subsymbols in $F_q$ and can obtain this bound with $n - k = q^m$. The second outcome of our research (Chapter 5) is devoted to repairing Reed-Solomon codes by heuristic algorithms where the repair schemes for Reed-Solomon codes employed in centralized storage systems and decentralized storage systems are proposed. The first goal of this chapter is carrying out a systematic study of Reed-Solomon codes used in centralized storage systems and investigate important aspects of repairing them under the trace repair framework, including which evaluation points to select and how to implement a trace repair scheme efficiently. In particular, we employ different heuristic algorithms to search for lowbandwidth repair schemes for codes of short lengths with typical redundancies and establish three tables of current best repair schemes for (n, k) Reed-Solomon codes over GF (256) with 4 ≤ n ≤ 16 and r = n - k ∈ {2, 3, 4} (Tables 5.1, 5.2, and 5.3). The tables cover most known codes currently used in the centralized-distributed storage industry. The second goal of Chapter 5 is to design compact repair groups that can tolerate as many failures as possible for the Reed-Solomon codes utilized in decentralized storage systems. It turns out that the maximal number of failures can be tolerated equals the size of a minimum hitting set minus one. When the repair groups for each symbol are generated from a single subspace (single seed), we establish a pair of asymptotically tight lower bound and upper bound on the size of such a minimum hitting set. Using Burnside’s Lemma and the Möbius inversion formula, we determine a number of subspaces that together attain the upper bound on the minimum hitting set size when the repair groups are generated from multiple subspaces (multiple seeds). We consider in Chapter 6, the last part of our research, repairing Reed-Solomon codes in the scenario in which some information of the lost symbol is known (the side information). The side information is represented as a set S of linearly independent combinations of the sub-symbols of the lost symbol. When S = ∅, this reduces to the standard repair problem of repairing a single codeword symbol. When S is a set of sub-symbols of an erased symbol, this becomes the repair problem with partially lost/erased symbol. We first establish that the minimum repair bandwidth depends on |S| and not the content of S and construct a lower bound on the repair bandwidth of a linear repair scheme with side information S. We then consider the well-known subspace-polynomial repair schemes and show that their repair bandwidths can be optimized by choosing the right subspaces. We also demonstrate several parameter regimes where the optimal bandwidths can be achieved for full-length Reed-Solomon codes. In the next part of this chapter, we consider constructing subspaces that can be applied to design the subspace polynomial repair schemes with low/optimal repair bandwidth.<p></p>