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, and compare very well in comparison to scheme that employs space-sharing between MSR and MBR codes. For the parameter set $(n,k,d=k)$ with $n < 2k-1$, we construct a class of codes with an auxiliary parameter $w$, referred to as canonical codes. With $w$ in the range $n-k < w < k$, these codes operate in the region between the MSR point and the MBR point, and perform significantly better than the space-sharing line. They only require a field size greater than $w+n-k$. For the case of $(n,n-1,n-1)$, canonical codes can also be shown to achieve an interior point on the line-segment joining the MSR point and the next point of slope-discontinuity on the storage-repair-bandwidth tradeoff. Thus we establish the existence of exact-repair codes on a point other than the MSR and the MBR point on the storage-repair-bandwidth tradeoff. We also construct layered regenerating codes for general parameter set $(n,k<d,k)$, which we refer to as non-canonical codes. These codes also perform significantly better than the space-sharing line, though they require a significantly higher field size. All the codes constructed in this paper are high-rate, can repair multiple node-failures and do not require any computation at the helper nodes. We also construct optimal codes with locality in which the local codes are layered regenerating codes.Comment: 20 pages, 9 figure

An Improved Outer Bound on the Storage-Repair-Bandwidth Tradeoff of Exact-Repair Regenerating Codes

In this paper we establish an improved outer bound on the storage-repair-bandwidth tradeoff of regenerating codes under exact repair. The result shows that in particular, it is not possible to construct exact-repair regenerating codes that asymptotically achieve the tradeoff that holds for functional repair. While this had been shown earlier by Tian for the special case of $[n,k,d]=[4,3,3]$ the present result holds for general $[n,k,d]$. The new outer bound is obtained by building on the framework established earlier by Shah et al.Comment: 14 page

Codes With Hierarchical Locality

In this paper, we study the notion of {\em codes with hierarchical locality} that is identified as another approach to local recovery from multiple erasures. The well-known class of {\em codes with locality} is said to possess hierarchical locality with a single level. In a {\em 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.Comment: 12 pages, submitted to ISIT 201

An Alternate Construction of an Access-Optimal Regenerating Code with Optimal Sub-Packetization Level

Given the scale of today's 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 $r^k$ 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 $r^{k/r}$ 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 $r^{k/r}$.Comment: To appear in National Conference on Communications 201

Private Balance-Checking on Blockchain Accounts Using Private Integer Addition

A transaction record in a sharded blockchain can be represented as a two-dimensional array of integers with row-index associated to an account, column-index to a shard and the entry to the transaction amount. In a blockchain-based cryptocurrency system with coded sharding, a transaction record of a given epoch of time is encoded using a block code considering the entries as finite-field symbols. Each column of the resultant coded array is then stored in a server. In the particular case of PolyShard scheme, the block code turns out to be a maximum-distance-separable code. In this paper, we propose a privacy-preserving multi-round protocol that allows a remote client to retrieve from a coded blockchain system the sum of transaction amounts belonging to two different epochs of time, but to the same account. At the core of the protocol lies an algorithm for a remote client to privately compute a non-linear function referred to as integer-addition of two finite-field symbols representing integer numbers, in the presence of curious-but-honest adversaries. Applying it to balance-checking in a cryptocurrency system, the protocol guarantees information-theoretic privacy on account number and shard number thereby ensuring perfect user anonymity, and also maintains confidentiality of half of the input bits on average. The protocol turns out to be a useful primitive for balance-checking in lightweight clients of a PolyShard-ed blockchain

On Maximally Recoverable Codes for Product Topologies

Given a topology of local parity-check constraints, a maximally recoverable code (MRC) can correct all erasure patterns that are information-theoretically correctable. In a grid-like topology, there are $a$ local constraints in every column forming a column code, $b$ local constraints in every row forming a row code, and $h$ global constraints in an $(m \times n)$ grid of codeword. Recently, Gopalan et al. initiated the study of MRCs under grid-like topology, and derived a necessary and sufficient condition, termed as the regularity condition, for an erasure pattern to be recoverable when $a=1, h=0$. In this paper, we consider MRCs for product topology ($h=0$). First, we construct a certain bipartite graph based on the erasure pattern satisfying the regularity condition for product topology (any $a, b$, $h=0$) and show that there exists a complete matching in this graph. We then present an alternate direct proof of the sufficient condition when $a=1, h=0$. We later extend our technique to study the topology for $a=2, h=0$, and characterize a subset of recoverable erasure patterns in that case. For both $a=1, 2$, our method of proof is uniform, i.e., by constructing tensor product $G_{\text{col}} \otimes G_{\text{row}}$ of generator matrices of column and row codes such that certain square sub-matrices retain full rank. The full-rank condition is proved by resorting to the matching identified earlier and also another set of matchings in erasure sub-patterns.Comment: 6 pages, accepted to National Conference of Communications (NCC) 201

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

In this paper, three outer bounds on the normalised storage-repair bandwidth trade-off of regenerating codes having parameter set {(n, k, d),(alpha, beta)} under the exact-repair (ER) setting are presented. The first outer bound, termed as the repair-matrix bound, is applicable for every parameter set (n, k, d), and in conjunction with a code construction known as improved layered codes, it characterises the normalised ER trade-off for the case (n, k = 3, d = n - 1). The bound shows that a non-vanishing gap exists between the ER and functional-repair (FR) trade-offs for every (n, k, d). The second bound, termed as the improved Mohajer-Tandon bound, is an improvement upon an existing bound due to Mohajer et al. and performs better in a region away from the minimum-storage-regenerating (MSR) point. However, in the vicinity of the MSR point, the repair-matrix bound outperforms the improved Mohajer-Tandon bound. The third bound is applicable to linear codes for the case k = d. In conjunction with the class of layered codes, the third outer bound characterises the normalised ER trade-off in the case of linear codes when k = d = n - 1