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June 2003 - February 2016
January 2003 - December 2012
January 1983 - May 2003
Education
August 1979 - August 1983
Publications
Publications (327)
In the first part of this two-part paper, we construct a family MFD$_2$ of low-correlation quaternary spreading codes having period $2046$. By quaternary, we mean that the spreading code symbols are drawn from $Z_4$ and are designed to be used in conjunction with QPSK modulation. Apart from low auto and crosscorrelation properties, we also require...
This paper presents an explicit construction of a class of optimal-access, minimum storage regenerating (MSR) codes, for small values of the number
d
of helper nodes. The construction is valid for any parameter set (
n, k, d
) with
d
∈ {
k
+1,
k
+2,
k
+3} and employs a finite field F
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xml...
For a convolutional code in the presence of a symbol erasure channel, the information debt $I(t)$ at time $t$ provides a measure of the number of additional code symbols required to recover all message symbols up to time $t$. Information-debt-optimal streaming ($i$DOS) codes are convolutional codes which allow for the recovery of all message symbol...
Global Navigation Satellite Systems (GNSS) employ low-correlation sequences, termed as spreading codes, to distinguish between the signals transmitted by the different satellites. The spreading codes commonly employed have period that is a multiple of 1023, as the fundamental frequency associated with the navigation signals generated onboard all of...
Streaming codes are a class of packet-level erasure codes that ensure packet recovery over a sliding window channel which allows either a burst erasure of size
b
or
a
random erasures within any window of size (τ + 1) time units, under a strict decoding-delay constraint τ. The field size over which streaming codes are constructed is an important...
We present two lower bounds on sub-packetization level
$\alpha $
of MSR codes with parameters
$(n, k, d=n-1, \alpha )$
where
$n$
is the block length,
$d$
is the number of helper nodes contacted during single-node repair,
$\alpha $
the sub-packetization level and
$k\alpha $
the scalar dimension. The first bound we present is for any MSR...
Data storage has grown such that distributed storage over a number of systems is now commonplace. This has given rise to an increase in the complexity of ensuring data loss does not occur, particularly where failure is due to the failure of individual nodes within the storage system. Redundancy was the main tool to combat this, but with huge increa...
Streaming codes are a class of packet-level erasure codes that are designed with the goal of ensuring recovery in low-latency fashion, of erased packets over a communication network. It is well-known in the streaming code literature, that diagonally embedding codewords of a $[\tau+1,\tau+1-a]$ Maximum Distance Separable (MDS) code within the packet...
Streaming codes are a class of packet-level erasure codes that ensure packet recovery over a sliding window channel which allows either a burst erasure of size $b$ or $a$ random erasures within any window of size $(\tau+1)$ time units, under a strict decoding-delay constraint $\tau$. The field size over which streaming codes are constructed is an i...
Streaming codes represent a packet-level FEC scheme for achieving reliable, low-latency communication. In the literature on streaming codes, the commonly-assumed Gilbert-Elliott channel model, is replaced by a more tractable, delay-constrained, sliding-window (DCSW) channel model that can introduce either random or burst erasures. The known streami...
This chapter deals with the topic of designing reliable and efficient codes for the storage and retrieval of large quantities of data over storage devices that are prone to failure. For long, the traditional objective has been one of ensuring reliability against data loss while minimizing storage overhead. More recently, a third concern has surface...
In this paper, we first provide analytical expressions for the block erasure probability (BEP) of block codes that can recover erasures that are (1) random with weight atmost $a$ or (2) burst with atmost $b$ consecutive erasures when used over Gilbert-Elliott (GE) channel. The same approach is then used to come up with tractable upper and lower bou...
An $(a,b,\tau)$ streaming code is a packet-level erasure code that can recover under a strict delay constraint of $\tau$ time units, from either a burst of $b$ erasures or else of $a$ random erasures, occurring within a sliding window of time duration $w$. While rate-optimal constructions of such streaming codes are available for all parameters $\{...
In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width w contains either up to a random erasures or an erasure burst of length at most b. One n...
This paper considers the natural extension of locally recoverable codes (LRC) to the case of t > 1 erased symbols. While several approaches have been proposed for the handling of multiple erasures, in the approach considered here, the t erased symbols are recovered in succession, each time contacting at most r other symbols for assistance. Under th...
Streaming codes offer reliable recovery under decoding-delay constraint
$\tau $
, of packets transmitted over a burst-and-random-erasure channel. Prior rate-optimal code constructions had field size quadratic in
$\tau $
and employed diagonal embedding of a scalar block code of length
$n$
within the packet stream. It is shown here that stagger...
Codes with locality are a class of codes introduced by Gopalan
et al.
to efficiently repair a failed node, by minimizing the number of nodes contacted during repair. An
$[n,k]$
systematic code is said to have information locality
$r$
, if each message symbol can be recovered by accessing
$\leq r$
other symbols. An
$[n,k]$
code is said to...
In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width $w$ contains either up to $a$ random erasures or an erasure burst of length at most $b$....
By a locally recoverable code (LRC), we will in this paper, mean a linear code in which a given code symbol can be recovered by taking a linear combination of at most $r$ other code symbols with $r << k$. A natural extension is to the local recovery of a set of $t$ erased symbols. There have been several approaches proposed for the handling of mult...
In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing t...
This thesis makes several significant contributions to the theory of both Regenerating (RG) and Locally Recoverable (LR) codes. The two principal contributions are characterizing the optimal rate of an LR code designed to recover from $t$ erased symbols sequentially, for any $t$ and the development of a tight bound on the sub-packetization level (l...
In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing t...
Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code ${\cal C}$, identify for each dimension $\nu$, the smallest size of the support of a subcode of ${\cal C}$ of dimension $\nu$. The...
In this paper, we study vector codes with all-symbol locality, where the local code is either a Minimum Bandwidth Regenerating (MBR) code or a Minimum Storage Regenerating (MSR) code. In the first part, we present vector codes with all-symbol MBR locality, for all parameters, that have both optimal minimum-distance and optimal rate. These codes com...
This paper presents the construction of an explicit, optimal-access, high-rate MSR code for any $(n,k,d=k+1,k+2,k+3)$ parameters over the finite field $\fQ$ having sub-packetization $\alpha = q^{\lceil\frac{n}{q}\rceil}$, where $q=d-k+1$ and $Q = O(n)$. The sub-packetization of the current construction meets the lower bound proven in a recent work...
In this survey, we present some fundamental concepts and theoretical advances attributable to Solomon Golomb together with the history and applications of this work to communications, coding and cryptography, along with some long-standing conjectures. Examples include the first engineering problem relating to feedback shift-register sequences, that...
In this paper, we show how the presence of locality within a binary cyclic code can be exploited to improve decoding performance and to reduce decoding complexity. We pursue two approaches. Under the first approach, we show how the Ordered Statistics Decoding (OSD) method can be modified by inserting a simple single round belief-propagation step at...
An erasure code is said to be a code with sequential recovery with parameters $r$ and $t$, if for any $s \leq t$ erased code symbols, there is an $s$-step recovery process in which at each step we recover exactly one erased code symbol by contacting at most $r$ other code symbols. In earlier work by the same authors, presented at ISIT 2017, we had...
Recovery of data packets from packet erasures in a timely manner is critical for many streaming applications. An early paper by Martinian and Sundberg introduced a framework for streaming codes and designed rate-optimal codes that permit delay-constrained recovery from an erasure burst of length up to $B$. A recent work by Badr et al. extended this...
Repair operations in erasure-coded distributed storage systems involves a lot of data movement. This can potentially expose data to malicious acts of passive eavesdroppers or active adversaries, putting security of the system at risk. This paper presents coding schemes and repair algorithms that ensure security of the data in the presence of passiv...
Let ${\cal C}$ be an $\{(n,k,d), (\alpha,\beta)\}$ linear, Minimum Storage Regenerating (MSR) code over the vector alphabet $\mathbb{F}_q^{\alpha}$ with linear repair of a failed node. Then $\mathcal{C}$ has the property that repair of the failure of any one single node (data or parity) can be carried out using linear operations, by uniformly downl...
The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with replicated-server PIR. In the present paper, the construction of an $(n,k)$ $\tau$-server binary, linear PIR code having parameters $n =...
In this paper, we study the impact of locality on the decoding of binary cyclic codes under two approaches, namely ordered statistics decoding (OSD) and trellis decoding. Given a binary cyclic code having locality or availability, we suitably modify the OSD to obtain gains in terms of the Signal-To-Noise ratio, for a given reliability and essential...
This paper presents an explicit construction for an $((n=2qt,k=2q(t-1),d=n-(q+1)), (\alpha = q(2q)^{t-1},\beta = \frac{\alpha}{q}))$ regenerating code over a field $\mathbb{F}_Q$ 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 $\alpha \...
An [n, k] code C is said to be locally recoverable in the presence of a single erasure, and with locality parameter r, if each of the n code symbols of C can be recovered by accessing at most r other code symbols. An [n, k] code is said to be a locally recoverable code with sequential recovery from t erasures, if for any set of s ≤ t erasures, ther...
An $[n,k]$ code $\mathcal{C}$ is said to be locally recoverable in the presence of a single erasure, and with locality parameter $r$, if each of the $n$ code symbols of $\mathcal{C}$ can be recovered by accessing at most $r$ other code symbols. An $[n,k]$ code is said to be a locally recoverable code with sequential recovery from $t$ erasures, if f...
In this paper we investigate bounds on rate and minimum distance of codes with $t$ availability. We present bounds on minimum distance of a code with $t$ availability that are tighter than existing bounds. For bounds on rate of a code with $t$ availability, we restrict ourselves to a sub-class of codes with $t$ availability called codes with strict...
This paper presents an explicit construction for an $((n,k,d), (\alpha,\beta))$ regenerating code over a field $\mathbb{F}_Q$ 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 given by $r^{\frac{n}{r}}$, for $r=(n-k)$, field size no larger than...
In this paper, codes with locality for four erasures are considered. An upper bound on the rate of codes with locality with sequential recovery from four erasures is derived. The rate bound derived here is field independent. An optimal construction for binary codes meeting this rate bound is also provided. The construction is based on regular graph...
In this paper, three outer bounds on the normalized storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set $\{(n,k,d),(\alpha,\beta)\}$ under the exact-repair (ER) setting are presented. The first outer bound is applicable for every parameter set $(n,k,d)$ and in conjunction with a code construction known as {\em improv...
In this paper, three outer bounds on the normalized storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set $\{(n,k,d),(\alpha,\beta)\}$ under the exact-repair (ER) setting are presented. The first outer bound is applicable for every parameter set $(n,k,d)$ and in conjunction with a code construction known as {\em improv...
This paper presents the development of a passive infra-red sensor tower platform along with a classification algorithm to distinguish between human intrusion, animal intrusion and clutter arising from wind-blown vegetative movement in an outdoor environment. The research was aimed at exploring the potential use of wireless sensor networks as an ear...
An early paper by Rashmi et. al. presented the construction of an $(n,k,d=n-1)$ MBR regenerating code featuring the inherent double replication of all code symbols and repair-by-transfer (RBT), both of which are important in practice. We first show that no MBR code can contain even a single code symbol that is replicated more than twice. We then go...
The focus of this paper is on linear, binary codes with locality having locality parameter $r$, that are capable of recovering from $t\geq 2$ erasures and that moreover, have short block length. Both sequential and parallel (through orthogonal parity checks) recovery is considered here. In the case of parallel repair, minimum-block-length construct...
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 ac...
The development of a Passive Infra-Red (PIR) sensing based intrusion detection system is presented here having the ability to reject vegetative clutter and distinguish between human and animal intrusions. This has potential application to reducing human-animal conflicts in the vicinity of a wildlife park. The system takes on the form of a sensor-to...
An [n, k] linear code C that is subject to locality constraints imposed by a
parity check matrix H0 is said to be a maximally recoverable (MR) code if it
can recover from any erasure pattern that some k-dimensional subcode of the
null space of H0 can recover from. The focus in this paper is on MR codes
constrained to have all-symbol locality r. Giv...
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...
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),...
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 t...
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 em...
Regenerating codes and codes with locality are two coding schemes that have recently been proposed, which in addition to ensuring data collection and reliability, also enable efficient node repair. In a situation where one is attempting to repair a failed node, regenerating codes seek to minimize the amount of data downloaded for node repair, while...
In this paper, we evaluate the efficacy, in a Hadoop setting, of two coding
schemes, both possessing an inherent double replication of data. The two coding
schemes belong to the class of regenerating and locally regenerating codes
respectively, and these two classes are representative of recent advances made
in designing codes for the efficient sto...
In this paper, we consider the security of exact-repair regenerating codes operating at the minimum-storage-regenerating (MSR) point. The security requirement (introduced in Shah et. al.) is that no information about the stored data file must be leaked in the presence of an eavesdropper who has access to the contents of ℓ1 nodes as well as all the...
In this paper, we study codes with locality in the presence of any two code
symbol erasures, such that both the erasures can be recovered by a sequence of
two local parity check computations. By a local parity check computation, we
mean recovery via a parity check equation associated with small Hamming weight.
These codes, which we refer to as loca...
It is our great pleasure to welcome you to COMSNETS 2014, the 6th International Conference on COMmunication Systems and NETworkS, and to Bangalore, India's technology city. COMSNETS is a premier international conference dedicated to advances in Networking and Communications Systems. The conference is a yearly event for a world-class gathering of re...