2,096 research outputs found

### 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

- …