Reed-Solomon (RS) codes are widely used in distributed storage systems. In this paper, we study the repair bandwidth and sub-packetization size of RS codes. The repair bandwidth is defined as the amount of transmitted information from surviving nodes to a failed node. The RS code can be viewed as a polynomial over a finite field GF (q ℓ ) evaluated at a set of points, where ℓ is called the sub-packetization size. Smaller bandwidth reduces the network traffic in distributed storage, and smaller ℓ facilitates the implementation of RS codes with lower complexity. Recently, Guruswami and Wootters proposed a repair method for RS codes when the evaluation points are the entire finite field. While the sub-packization size can be arbitrarily small, the repair bandwidth is higher than the minimum storage regenerating (MSR) bound. Tamo, Ye and Barg achieved the MSR bound but the sub-packetization size grows faster than the exponential function of the number of the evaluation points. In this work, we present code constructions and repair schemes that extend these results to accommodate different sizes of the evaluation points. In other words, we design schemes that provide points in between. These schemes provide a flexible tradeoff between the sub-packetization size and the repair bandwidth. In addition, we generalize our schemes to manage multiple failures.

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This paper presents flexible storage codes, a class of error-correcting codes that can recover information from a flexible number of storage nodes. As a result, one can make better use of the available storage nodes in the presence of unpredictable node failures and reduce the data access latency. Assume a storage system encodes kℓ information symbols over a finite field F into n nodes, each of size ℓ symbols. The code is parameterized by a set of tuples {(Rj, ℓj) : 1 ≤ j ≤ a}, satisfying ℓ1 < ℓ2 < ... < ℓa = ℓ and R1 > R2 > • • • > Ra, such that the information symbols can be reconstructed from any Rj nodes, each node accessing ℓj symbols, for any 1 ≤ j ≤ a. In other words, the code allows a flexible number of nodes for decoding to accommodate the variance in the data access time of the nodes. Code constructions are presented for different storage scenarios, including LRC (locally recoverable) codes, PMDS (partial MDS) codes, and MSR (minimum storage regenerating) codes. We analyze the latency of accessing information and perform simulations on Amazon clusters to show the efficiency of the presented codes.