arXiv subjects

P. Vijay Kumar

Publications and source records attributed to P. Vijay Kumar.

At least 19 recordsLinked to original sources

arXiv2025

Latency-Optimal File Assignment in Geo-Distributed Storage with Preferential Demands

We consider the problem of data storage in a geographically distributed (or geo-distributed) network of servers (or nodes) where inter-node communication incurs certain round-trip delays. Every node serves a set of users who can request any file in the network. If the requested file is not available at the node, it communicates with other nodes to obtain the file, thus causing the user to experience latency in obtaining the file. The files can be placed uncoded, where each node stores exact copies of the files, or in coded fashion, where certain linear combination of files are placed at each node. We aim to obtain an optimal file placement on the nodes with respect to minimizing the worst-case latency at each node, as well as the system-average latency. The prior literature considered the case of equiprobable file demands at the nodes. In this paper, we investigate the generic case of non-uniform file-demand probabilities at each node. The scheme presented here is optimal within the family of uncoded schemes. It is obtained first by modeling the worst-case latency constraint as a vertex coloring problem, and then converting the system-average latency optimization to a problem of balanced-assignment.

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arXiv2025

On the Efficacy of the Peeling Decoder for the Quantum Expander Code

The problem of recovering from qubit erasures has recently gained attention as erasures occur in many physical systems such as photonic systems, trapped ions, superconducting qubits and circuit quantum electrodynamics. While several linear-time decoders for error correction are known, their error-correcting capability is limited to half the minimum distance of the code, whereas erasure correction allows one to go beyond this limit. As in the classical case, stopping sets pose a major challenge in designing efficient erasure decoders for quantum LDPC codes. In this paper, we show through simulation, that an attractive alternative here, is the use of quantum expander codes in conjunction with the peeling decoder that has linear complexity. We also discuss additional techniques including small-set-flip decoding, that can be applied following the peeling operation, to improve decoding performance and their associated complexity.

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arXiv2024

Quaternary and Component-Binary Spreading Codes with Low Correlation for Navigation Systems

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 in addition, to our knowledge for the first time, that the spreading code family IZ4$_2$ obtained by taking the union of the component in-phase and quadrature-phase binary spreading codes associated to each quaternary spreading code in MFD$_2$, also have desirable low-correlation properties. We also investigate the balance of the quaternary and binary spreading codes. The second part is motivated by an application to the design of spreading code, (in this application termed as ranging codes), having parameters suitable for use in a lunar PNT system. Two lengths that are of particular current interest for a planned lunar PNT satellite system are $2046$ and $10230$. We study the applicability of a subset IZ4$_{2S}$ of IZ4$_2$ containing balanced binary spreading codes having length $2046$ to such a lunar PNT system. We show that the spreading codes belonging to IZ4$_{2S}$ compare favorably with the spreading codes of length $2046$ appearing in a recent issue of Inside GNSS. We also show that the IZ4$_{10}$ spreading code family in which the spreading codes have length $10230$, compares well in comparison with spreading codes of length $10230$ described in this article. In addition, the IZ4$_{10}$ and IZ4$_2$ spreading codes have been paired so as to be orthogonal at zero shift despite their different lengths and chipping rates.

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arXiv2024

On Existence of Latency Optimal Uncoded Storage Schemes in Geo-Distributed Data Storage Systems

We consider the problem of geographically distributed data storage in a network of servers (or nodes) where the nodes are connected to each other via communication links having certain round-trip times (RTTs). Each node serves a specific set of clients, where a client can request for any of the files available in the distributed system. The parent node provides the requested file if available locally; else it contacts other nodes that have the data needed to retrieve the requested file. This inter-node communication incurs a delay resulting in a certain latency in servicing the data request. The worst-case latency incurred at a servicing node and the system average latency are important performance metrics of a storage system, which depend not only on inter-node RTTs, but also on how the data is stored across the nodes. Data files could be placed in the nodes as they are, i.e., in uncoded fashion, or can be coded and placed. This paper provides the necessary and sufficient conditions for the existence of uncoded storage schemes that are optimal in terms of both per-node worst-case latency and system average latency. In addition, the paper provides efficient binary storage codes for a specific case where optimal uncoded schemes do not exist.

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arXiv2024

On Streaming Codes for Burst and Random Errors

Streaming codes (SCs) are packet-level codes that recover erased packets within a strict decoding-delay deadline. Streaming codes for various packet erasure channel models such as sliding-window (SW) channel models that admit random or burst erasures in any SW of a fixed length have been studied in the literature, and the optimal rate as well as rate-optimal code constructions of SCs over such channel models are known. In this paper, we study error-correcting streaming codes ($\text{SC}_{\text{ERR}}$s), i.e., packet-level codes which recover erroneous packets within a delay constraint. We study $\text{SC}_{\text{ERR}}$s for two classes of SW channel models, one that admits random packet errors, and another that admits multiple bursts of packet errors, in any SW of a fixed length. For the case of random packet errors, we establish the equivalence of an $\text{SC}_{\text{ERR}}$ and a corresponding SC that recovers from random packet erasures, thus determining the optimal rate of an $\text{SC}_{\text{ERR}}$ for this setting, and providing a rate-optimal code construction for all parameters. We then focus on SCs that recover from multiple erasure bursts and derive a rate-upper-bound for such SCs. We show the necessity of a divisibility constraint for the existence of an SC constructed by the popular diagonal embedding technique, that achieves this rate-bound under a stringent delay requirement. We then show that a construction known in the literature achieves this rate-bound when the divisibility constraint is met. We further show the equivalence of the SCs considered and $\text{SC}_{\text{ERR}}$s for the setting of multiple error bursts, under a stringent delay requirement.

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arXiv2024

On Streaming Codes for Simultaneously Correcting Burst and Random Erasures

Streaming codes are packet-level codes that recover dropped packets within a strict decoding-delay constraint. We study streaming codes over a sliding-window (SW) channel model which admits only those erasure patterns which allow either a single burst erasure of $\le b$ packets along with $\le e$ random packet erasures, or else, $\le a$ random packet erasures, in any sliding-window of $w$ time slots. We determine the optimal rate of a streaming code constructed via the popular diagonal embedding (DE) technique over such a SW channel under delay constraint $τ=(w-1)$ and provide an $O(w)$ field size code construction. For the case $e>1$, we show that it is not possible to significantly reduce this field size requirement, assuming the well-known MDS conjecture. We then provide a block code construction whose DE yields a streaming code achieving the rate derived above, over a field of size sub-linear in $w,$ for a family of parameters having $e=1.$ We show the field size optimality of this construction for some parameters, and near-optimality for others under a sparsity constraint. Additionally, we derive an upper-bound on the $d_{\text{min}}$ of a cyclic code and characterize cyclic codes which achieve this bound via their ability to simultaneously recover from burst and random erasures.

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arXiv2023

Explicit Information-Debt-Optimal Streaming Codes With Small Memory

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 symbols up to $t$ whenever $I(t)$ turns zero under the following conditions; (i) information debt can be non-zero for at most $τ$ consecutive time slots and (ii) information debt never increases beyond a particular threshold. The existence of periodically-time-varying $i$DOS codes are known for all parameters. In this paper, we address the problem of constructing explicit, time-invariant $i$DOS codes. We present an explicit time-invariant construction of $i$DOS codes for the unit memory ($m=1$) case. It is also shown that a construction method for convolutional codes due to Almeida et al. leads to explicit time-invariant $i$DOS codes for all parameters. However, this general construction requires a larger field size than the first construction for the $m=1$ case.

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arXiv2022

Rate-Optimal Streaming Codes Over the Three-Node Decode-And-Forward Relay Network

In this paper, we study the three-node Decode-and-Forward (D&F) relay network subject to random and burst packet erasures. The source wishes to transmit an infinite stream of packets to the destination via the relay. The three-node D&F relay network is constrained by a decoding delay of T packets, i.e., the packet transmitted by the source at time i must be decoded by the destination by time i+T. For the individual channels from source to relay and relay to destination, we assume a delay-constrained sliding-window (DCSW) based packet-erasure model that can be viewed as a tractable approximation to the commonly-accepted Gilbert-Elliot channel model. Under the model, any time-window of width w contains either up to a random erasure or else erasure burst of length at most b (>= a). Thus the source-relay and relay-destination channels are modeled as (a_1, b_1, w_1, T_1) and (a_2, b_2, w_2, T_2) DCSW channels. We first derive an upper bound on the capacity of the three-node D&F relay network. We then show that the upper bound is tight for the parameter regime: max{b_1, b_2}|(T-b_1-b_2-max{a_1, a_2}+1), a1=a2 OR b1=b2 by constructing streaming codes achieving the bound. The code construction requires field size linear in T, and has decoding complexity equivalent to that of decoding an MDS code.

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arXiv2022

An Explicit, Coupled-Layer Construction of a High-Rate Regenerating Code with Low Sub-Packetization Level, Small Field Size and $d< (n-1)$

This paper presents an explicit construction for an $((n=2qt,k=2q(t-1),d=n-(q+1)), (α= q(2q)^{t-1},β= \fracα{q}))$ regenerating code (RGC) over a field $\mathbb{F}_Q$ having rate $\geq \frac{t-2}{t}$. The RGC code can be constructed to have rate $k/n$ as close to $1$ as desired, sub-packetization level $α\leq r^{\frac{n}{r}}$ for $r=(n-k)$, field size $Q$ no larger than $n$ and where all code symbols can be repaired with the same minimum data download.

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arXiv2021

Small-d MSR Codes with Optimal Access, Optimal Sub-Packetization and Linear Field Size

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 $d \in \{k+1, k+2, k+3\}$ and employs a finite field $\mathbb{F}_q$ of size $q=O(n)$. We will refer to the constructed codes as Small-d MSR codes. The sub-packetization level $α$ is given by $α= s^{{\lceil\frac{n}{s}\rceil}}$, where $s=d-k+1$. By an earlier result on the sub-packetization level for optimal-access MSR codes, this is the smallest value possible.

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arXiv2021

On Lower Bounds on Sub-Packetization Level of MSR codes and On The Structure of Optimal-Access MSR Codes Achieving The Bound

We present two lower bounds on sub-packetization level $α$ of MSR codes with parameters $(n, k, d=n-1, α)$ where $n$ is the block length, $k$ dimension, $d$ number of helper nodes contacted during single node repair and $α$ the sub-packetization level. The first bound we present is for any MSR code and is given by $α\ge e^{\frac{(k-1)(r-1)}{2r^2}}$. The second bound we present is for the case of optimal-access MSR codes and the bound is given by $α\ge \min \{ r^{\frac{n-1}{r}}, r^{k-1} \}$. There exist optimal-access MSR constructions that achieve the second sub-packetization level bound with an equality making this bound tight. We also prove that for an optimal-access MSR codes to have optimal sub-packetization level under the constraint that the indices of helper symbols are dependant only on the failed node, it is needed that the support of the parity check matrix is same as the support structure of several other optimal constructions in literature.

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arXiv2021

On the Performance Analysis of Streaming Codes over the Gilbert-Elliott Channel

The Gilbert-Elliot (GE) channel is a commonly-accepted model for packet erasures in networks. Streaming codes are a class of packet-level erasure codes designed to provide reliable communication over the GE channel. The design of a streaming code may be viewed as a two-step process. In the first, a more tractable, delay-constrained sliding window (DCSW) channel model is considered as a proxy to the GE channel. The streaming code is then designed to reliably recover from all erasures introduced by the DCSW channel model. Simulation is typically used to evaluate the performance of the streaming code over the original GE channel, as analytic performance evaluation is challenging. In the present paper, we take an important first step towards analytical performance evaluation. Recognizing that most, efficient constructions of a streaming code are based on the diagonal embedding or horizontal embedding of scalar block codes within a packet stream, this paper provides upper and lower bounds on the block-erasure probability of the underlying scalar block code when operated over the GE channel.

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arXiv2021

Locally Recoverable Streaming Codes for Packet-Erasure Recovery

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 $[τ+1,τ+1-a]$ Maximum Distance Separable (MDS) code within the packet stream, leads to rate-optimal streaming codes capable of recovering from $a$ arbitrary packet erasures, under a strict decoding delay constraint $τ$. Thus MDS codes are geared towards the efficient handling of the worst-case scenario corresponding to the occurrence of $a$ erasures. In the present paper, we have an increased focus on the efficient handling of the most-frequent erasure patterns. We study streaming codes which in addition to recovering from $a>1$ arbitrary packet erasures under a decoding delay $τ$, have the ability to handle the more common occurrence of a single-packet erasure, while incurring smaller delay $r<τ$. We term these codes as $(a,τ,r)$ locally recoverable streaming codes (LRSCs), since our single-erasure recovery requirement is similar to the requirement of locality in a coded distributed storage system. We characterize the maximum possible rate of an LRSC by presenting rate-optimal constructions for all possible parameters $\{a,τ,r\}$. Although the rate-optimal LRSC construction provided in this paper requires large field size, the construction is explicit. It is also shown that our $(a,τ=a(r+1)-1,r)$ LRSC construction provides the additional guarantee of recovery from the erasure of $h, 1 \leq h \leq a$, packets, with delay $h(r+1)-1$. The construction thus offers graceful degradation in decoding delay with increasing number of erasures.

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arXiv2021

Explicit Rate-Optimal Streaming Codes with Smaller Field Size

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 factor determining the complexity of implementation. The best known explicit rate-optimal streaming code requires a field size of $q^2$ where $q \ge τ+b-a$ is a prime power. In this work, we present an explicit rate-optimal streaming code, for all possible $\{a,b,τ\}$ parameters, over a field of size $q^2$ for prime power $q \ge τ$. This is the smallest-known field size of a general explicit rate-optimal construction that covers all $\{a,b,τ\}$ parameter sets. We achieve this by modifying the non-explicit code construction due to Krishnan et al. to make it explicit, without change in field size.

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arXiv2021

Generalized Simple Streaming Codes from MDS Codes

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 streaming codes that are rate optimal over the DCSW channel model are constructed by diagonally embedding a scalar block code across successive packets. These code constructions have field size that is quadratic in the delay parameter $τ$ and have a somewhat complex structure with an involved decoding procedure. This led to the introduction of simple streaming (SS) codes in which diagonal embedding is replaced by staggered-diagonal embedding (SDE). The SDE approach reduces the impact of a burst of erasures and makes it possible to construct near-rate-optimal streaming codes using Maximum Distance Separable (MDS) code having linear field size. The present paper takes this development one step further, by retaining the staggered-diagonal feature, but permitting the placement of more than one code symbol from a given scalar codeword within each packet. These generalized, simple streaming codes allow us to improve upon the rate of SS codes, while retaining the simplicity of working with MDS codes. We characterize the maximum code rate of streaming codes under a constraint on the number of contiguous packets over which symbols of the underlying scalar code are dispersed. Such a constraint leads to simplified code construction and reduced-complexity decoding.

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arXiv2020

Codes for Distributed Storage

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 surfaced, namely of the need to efficiently recover from the failure of a single storage unit, corresponding to recovery from the erasure of a single code symbol. We explain here, how coding theory has evolved to tackle this fresh challenge.

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arXiv2020

Staggered Diagonal Embedding Based Linear Field Size Streaming Codes

An $(a,b,τ)$ streaming code is a packet-level erasure code that can recover under a strict delay constraint of $τ$ 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 $\{a,b,τ,w\}$ in the literature, they require in most instances, a quadratic, $O(τ^2)$ field size. In this work, we make further progress towards field size reduction and present rate-optimal $O(τ)$ field size streaming codes for two regimes: (i) $gcd(b,τ+1-a)\ge a$ (ii) $τ+1 \ge a+b$ and $b \mod \ a \in \{0,a-1\}$.

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arXiv2019

Low Field-size, Rate-Optimal Streaming Codes for Channels With Burst and Random Erasures

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 needs to recover any erased packet, where erasures are as per the channel model, with a strict decoding delay deadline of $τ$ time slots. Presently existing rate-optimal constructions in the literature require, in general, a field-size which grows exponential in $τ$, for a constant $\frac{a}τ$. In this work, we present a new rate-optimal code construction covering all channel and delay parameters, which requires an $O(τ^2)$ field-size. As a special case, when $(b-a)=1$, we have a field-size linear in $τ$. We also present three other constructions having linear field-size, under certain constraints on channel and decoding delay parameters. As a corollary, we obtain low field-size, rate-optimal convolutional codes for any given column distance and column span. Simulations indicate that the newly proposed streaming code constructions offer lower packet-loss probabilities compared to existing schemes, for selected instances of Gilbert-Elliott and Fritchman channels.

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