An Explicit Construction of Systematic MDS Codes with Small Sub-packetization for All-Node Repair

An explicit construction of systematic MDS codes, called HashTag+ codes, with arbitrary sub-packetization level for all-node repair is proposed. It is shown that even for small sub-packetization levels, HashTag+ codes achieve the optimal MSR point for repair of any parity node, while the repair bandwidth for a single systematic node depends on the sub-packetization level. Compared to other codes in the literature, HashTag+ codes provide from 20% to 40% savings in the average amount of data accessed and transferred during repair

New Centralized MSR Codes With Small Sub-packetization

Centralized repair refers to repairing $h\geq 2$ node failures using $d$ helper nodes in a centralized way, where the repair bandwidth is counted by the total amount of data downloaded from the helper nodes. A centralized MSR code is an MDS array code with $(h,d)$-optimal repair for some $h$ and $d$. In this paper, we present several classes of centralized MSR codes with small sub-packetization. At first, we construct an alternative MSR code with $(1,d_i)$-optimal repair for multiple repair degrees $d_i$ simultaneously. Based on the code structure, we are able to construct a centralized MSR code with $(h_i,d_i)$-optimal repair property for all possible $(h_i,d_i)$ with $h_i\mid (d_i-k)$ simultaneously. The sub-packetization is no more than ${\rm lcm}(1,2,\ldots,n-k)(n-k)^n$, which is much smaller than a previous work given by Ye and Barg ($({\rm lcm}(1,2,\ldots,n-k))^n$). Moreover, for general parameters $2\leq h\leq n-k$ and $k\leq d\leq n-h$, we further give a centralized MSR code enabling $(h,d)$-optimal repair with sub-packetization smaller than all previous works

MDS Array Codes With (Near) Optimal Repair Bandwidth for All Admissible Repair Degrees

Abundant high-rate (n, k) minimum storage regenerating (MSR) codes have been reported in the literature. However, most of them require contacting all the surviving nodes during a node repair process, resulting in a repair degree of d=n-1. In practical systems, it may not always be feasible to connect and download data from all surviving nodes, as some nodes may be unavailable. Therefore, there is a need for MSR code constructions with a repair degree of d<n-1. Up to now, only a few (n, k) MSR code constructions with repair degree d<n-1 have been reported, some have a large sub-packetization level, a large finite field, or restrictions on the repair degree d. In this paper, we propose a new (n, k) MSR code construction that works for any repair degree d>k, and has a smaller sub-packetization level or finite field than some existing constructions. Additionally, in conjunction with a previous generic transformation to reduce the sub-packetization level, we obtain an MDS array code with a small sub-packetization level and $(1+\epsilon)$-optimal repair bandwidth (i.e., $(1+\epsilon)$ times the optimal repair bandwidth) for repair degree d=n-1. This code outperforms some existing ones in terms of either the sub-packetization level or the field size.Comment: Submitted to the IEEE Transactions on Communication

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