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

Low Correlation Sequences over the QAM Constellation

This paper presents the first concerted look at low correlation sequence families over QAM constellations of size M^2=4^m and their potential applicability as spreading sequences in a CDMA setting. Five constructions are presented, and it is shown how such sequence families have the ability to transport a larger amount of data as well as enable variable-rate signalling on the reverse link. Canonical family CQ has period N, normalized maximum-correlation parameter theta_max bounded above by A sqrt(N), where 'A' ranges from 1.8 in the 16-QAM case to 3.0 for large M. In a CDMA setting, each user is enabled to transfer 2m bits of data per period of the spreading sequence which can be increased to 3m bits of data by halving the size of the sequence family. The technique used to construct CQ is easily extended to produce larger sequence families and an example is provided. Selected family SQ has a lower value of theta_max but permits only (m+1)-bit data modulation. The interleaved 16-QAM sequence family IQ has theta_max <= sqrt(2) sqrt(N) and supports 3-bit data modulation. The remaining two families are over a quadrature-PAM (Q-PAM) subset of size 2M of the M^2-QAM constellation. Family P has a lower value of theta_max in comparison with Family SQ, while still permitting (m+1)-bit data modulation. Interleaved family IP, over the 8-ary Q-PAM constellation, permits 3-bit data modulation and interestingly, achieves the Welch lower bound on theta_max.Comment: 21 pages, 3 figures. To appear in IEEE Transactions on Information Theory in February 200

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

A Tight Lower Bound on the Sub-Packetization Level of Optimal-Access MSR and MDS Codes

The first focus of the present paper, is on lower bounds on the sub-packetization level $\alpha$ of an MSR code that is capable of carrying out repair in help-by-transfer fashion (also called optimal-access property). We prove here a lower bound on $\alpha$ which is shown to be tight for the case $d=(n-1)$ by comparing with recent code constructions in the literature. We also extend our results to an $[n,k]$ MDS code over the vector alphabet. Our objective even here, is on lower bounds on the sub-packetization level $\alpha$ of an MDS code that can carry out repair of any node in a subset of $w$ nodes, $1 \leq w \leq (n-1)$ where each node is repaired (linear repair) by help-by-transfer with minimum repair bandwidth. We prove a lower bound on $\alpha$ for the case of $d=(n-1)$. This bound holds for any $w (\leq n-1)$ and is shown to be tight, again by comparing with recent code constructions in the literature. Also provided, are bounds for the case $d<(n-1)$. We study the form of a vector MDS code having the property that we can repair failed nodes belonging to a fixed set of $Q$ nodes with minimum repair bandwidth and in optimal-access fashion, and which achieve our lower bound on sub-packetization level $\alpha$. It turns out interestingly, that such a code must necessarily have a coupled-layer structure, similar to that of the Ye-Barg code.Comment: Revised for ISIT 2018 submissio

Codes with Locality for Two Erasures

In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated to small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan \textit{et al}, in which recovery from a single erasure is considered. By studying the Generalized Hamming Weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Tur\'an graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from $2$ erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.Comment: 14 pages, 3 figures, Updated for improved readabilit