Birenjith Sasidharan

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

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 space-sharing 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.

An improved outer bound on the storage-repair-bandwidth tradeoff of exact-repair regenerating codes

While the tradeoff between the amount of data stored and the repair bandwidth of an (n, k, d) regenerating code has been characterized under functional repair (FR), the case of exact repair (ER) remains unresolved. It is known that there do not exist ER codes which lie on the FR tradeoff at most of the points. The question as to whether one can asymptotically approach the FR tradeoff was settled recently by Tian who showed that in the (4, 3, 3) case, the ER region is bounded away from the FR region. The FR tradeoff serves as a trivial outer bound on the ER tradeoff. In this paper, we extend Tians results by establishing an improved outer bound on the ER tradeoff which shows that the ER region is bounded away from the FR region, for any (n, k, d). Our approach is analytical and builds upon the framework introduced earlier by Shah et. al. Interestingly, a recently-constructed, layered regenerating code is shown to achieve a point on this outer bound for the (5, 4, 4) case. This represents the first-known instance of an optimal ER code that does not correspond to a point on the FR tradeoff.

A high-rate MSR code with polynomial sub-packetization level

We present a high-rate (n, k, d = n - 1)-MSR code with a sub-packetization level that is polynomial in the dimension k of the code. While polynomial sub-packetization level was achieved earlier for vector MDS codes that repair systematic nodes optimally, no such MSR code construction is known. In the low-rate regime (i. e., rates less than one-half), MSR code constructions with a linear sub-packetization level are available. But in the high-rate regime (i. e., rates greater than one-half), the known MSR code constructions required a sub-packetization level that is exponential in k. In the present paper, we construct an MSR code for d = n - 1 with a fixed rate equation, achieveing a sub-packetization level α = O(kt). The code allows help-by-transfer repair, i. e., no computations are needed at the helper nodes during repair of a failed node.

An alternate construction of an access-optimal regenerating code with optimal sub-packetization level

Given the scale of todays distributed storage systems, the failure of an individual node is a common phenomenon. Various metrics have been proposed to measure the efficacy of the repair of a failed node, such as the amount of data download needed to repair (also known as the repair bandwidth), the amount of data accessed at the helper nodes, and the number of helper nodes contacted. Clearly, the amount of data accessed can never be smaller than the repair bandwidth. In the case of a help-by-transfer code, the amount of data accessed is equal to the repair bandwidth. It follows that a help-by-transfer code possessing optimal repair bandwidth is access optimal. The focus of the present paper is on help-by-transfer codes that employ minimum possible bandwidth to repair the systematic nodes and are thus access optimal for the repair of a systematic node. The zigzag construction by Tamo et al. in which both systematic and parity nodes are repaired is access optimal. But the sub-packetization level required is rk where r is the number of parities and k is the number of systematic nodes. To date, the best known achievable sub-packetization level for access-optimal codes is rk/r in a MISER-code-based construction by Cadambe et al. in which only the systematic nodes are repaired and where the location of symbols transmitted by a helper node depends only on the failed node and is the same for all helper nodes. Under this set-up, it turns out that this sub-packetization level cannot be improved upon. In the present paper, we present an alternate construction under the same setup, of an access-optimal code repairing systematic nodes, that is inspired by the zigzag code construction and that also achieves a sub-packetization level of rk/r.

High-rate regenerating codes through layering

In this paper, we provide explicit constructions for a class of exact-repair regenerating codes that possess a layered structure. These regenerating codes correspond to interior points on the storage-repair-bandwidth tradeoff where the cut-set bound of network coding is known to be not achievable under exact repair. The codes presented in this paper compare very well in comparison to schemes that employ space-sharing between MSR and MBR points, and come closest of all-known explicit constructions to interior points of the tradeoff. The codes can be constructed for a wide range of parameters, are high-rate, can repair multiple nodes simultaneously and no computation at helper nodes is required to repair a failed node. We also construct optimal codes with locality in which the local codes are layered regenerating codes.

An explicit, coupled-layer construction of a high-rate MSR code with low sub-packetization level, small field size and d < (n − 1)

This paper presents an explicit construction for an ((n = 2qt, k = 2q{t−1), d = n − (q + 1)), (α = q(2q)^t−1,β = α/q)) regenerating code over a field F<inf>q</inf> operating at the Minimum Storage Regeneration (MSR) point. The MSR code can be constructed to have rate k/n as close to 1 as desired, sub-packetization level α ≤ r^n/r for r = (n − k), field size Q no larger than n and where all code symbols can be repaired with the same minimum data download. This is the first-known construction of such an MSR code for d < (n − 1).

Codes with hierarchical locality

In this paper, we study the notion of codes with hierarchical locality that is identified as another approach to local recovery from multiple erasures. The well-known class of codes with locality is said to possess hierarchical locality with a single level. In a code with two-level hierarchical locality, every symbol is protected by an inner-most local code, and another middle-level code of larger dimension containing the local code. We first consider codes with two levels of hierarchical locality, derive an upper bound on the minimum distance, and provide optimal code constructions of low field-size under certain parameter sets. Subsequently, we generalize both the bound and the constructions to hierarchical locality of arbitrary levels.

Improved layered regenerating codes characterizing the exact-repair storage-repair bandwidth tradeoff for certain parameter sets

The characterization of the storage-repair bandwidth tradeoff of (n, k, d)-regenerating codes under the exact-repair setting remains an open problem. The problem has been solved only for the special case of (n, k, d) = (4, 3, 3). In the present paper, we characterize the tradeoff for the larger family of parameters (n, k = 3, d = n - 1). This is accomplished by constructing an (n, k <; d, d)-regenerating code, referred to as the improved layered code. In the case when (n, k = 3, d = n - 1), the code operates on a point that coincides with an interior point of a recently derived outer bound on the tradeoff. The code also achieves an interior point on the outer bound for the parameter set (n, k = 4, d = n - 1).

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

In this paper, three outer bounds on the normalized storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set

Erasure coding for distributed storage: an overview

\{(n,k,d),(\alpha,\beta)\}