Layered Exact-Repair Regenerating Codes via Embedded Error Correction and Block Designs

Abstract

A new class of exact-repair regenerating codes is constructed by stitching together shorter erasure correction codes, where the stitching pattern can be viewed as block designs. The proposed codes have the help-by-transfer property where the helper nodes simply transfer part of the stored data directly, without performing any computation. This embedded error correction structure makes the decoding process straightforward, and in some cases the complexity is very low. We show that this construction is able to achieve performance better than spacesharing between the minimum storage regenerating codes and the minimum repair-bandwidth regenerating codes, and it is the first class of codes to achieve this performance. In fact, it is shown that the proposed construction can achieve a nontrivial point on the optimal functional-repair tradeoff, and it is asymptotically optimal at high rate, i.e., it asymptotically approaches the minimum storage and the minimum repair-bandwidth simultaneously.

Outer bounds on the storage-repair bandwidth trade-off of exact-repair regenerating codes

“…However for the case of linear codes, the bound can be computed to obtain an explicit expression for any parameter set (n, k, d). A second paper by Tian, [20], characterizes the ER tradeoff for (n = 5, k = 4, d = 4) with the help of a class of codes known as the layered codes introduced in [21]. A different approach adopted to derive an outer bound on the normalized ER tradeoff is presented in [22].…”

“…Recently in [20], Tian made further progress with his computational approach to provide an upper bound on the ER file size for (n, k, d) = (5, 4, 4). In both the case of (4, 3, 3) and (5, 4, 4), the bounds are achieved using the well-known class of layered codes [21]. These results are made part of the online collection of "Solutions of Computed Information Theoretic Limits (SCITL)" hosted at [26].…”

“…In the case of linear codes, the bound presented in Theorem I.2 is achieved by canonical layered codes that was introduced in [21]. When specialized to the case of d = n − 1, the layered codes achieve points described by…”

A connection between locally repairable codes and exact regenerating codes

“…Results in [12] and [13] are combined in [16]. However, in contrast to these constructions we are not trying to find codes that perform as well as possible.…”

“…Example 4.1: Suppose (N, K, D, R, ∆) = (24,16,4,3,2). Now, R + 1 divides N so there are plenty of constructions showing that such code exists, see for example [6].…”