arXiv subjects

Myna Vajha

Publications and source records attributed to Myna Vajha.

18 recordsLinked to original sources

arXiv2026

Link Adaptation Using Joint-Thompson Sampling

The choice of Modulation and Coding (MCS) type for a particular channel condition is made through link adaptation (LA) algorithms that operate at the MAC layer. These algorithms rely on the ACK/NACK statistics and the channel quality index (CQI) feedback. Several existing works model LA as a multi-armed bandit (MAB) problem across cellular and Wi-Fi links. In the MAB formulation, each available MCS is a Bernoulli arm parameterized by its transmission success probability, and the goal is to design a selection strategy that accrues maximum reward. Several popular MAB algorithms, such as upper confidence bound (UCB) and Thompson Sampling (TS), have been proposed in the literature. Using the fact that MCS success probabilities are ordered, we propose the Joint-Thompson Sampling (Joint-TS) algorithm. Unlike classical TS, which assumes independent Beta distributions for each arm, Joint-TS utilizes a multivariate ordered Beta distribution as the prior to preserve the inherent monotonicity of success probabilities. Our simulation results show that while existing MAB algorithms fail in specific scenarios, Joint-TS delivers competitive throughput with robust, consistent performance in all scenarios.

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arXiv2026

Extreme Points of the $(0,\delta)$-LDP Polytope with Small Input Size and Arbitrary Output Sizes

The structure of locally differentially private (LDP) mechanisms can be understood through the geometry of the corresponding privacy polytope. While the extreme points of the $ (\epsilon,0)$-LDP polytope are well characterized (Kairouz \emph{et al.}, 2014; Holohan \emph{et al.}, 2017; Pensia \emph{et al.}, 2017), comparatively little is known for the $(\epsilon,\delta)$-LDP polytope with $\delta>0$. Recent work (Elangovan and Jog, 2024) has shown that even in the special case $\epsilon=0$, the $ (0,\delta) $-LDP privacy polytope exhibits fundamentally different behaviour. In this work, we provide complete characterizations of the extreme points for the low-input-alphabet regime $k=2$ and $k=3$ and with arbitrary output alphabet size $m $. We also identify new extreme mechanisms for larger input alphabet sizes $k$, of the star configuration type, as introduced by Elangovan and Jog (2024).

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arXiv2025

On MDS Convertible Codes in the Merge Regime

In large-scale distributed storage systems, erasure coding is employed to ensure reliability against disk failures. Recent work by Kadekodi et al. demonstrates that adapting code parameters to varying disk failure rates can lead to significant storage savings without compromising reliability. Such adaptations, known as \emph{code conversions}, motivate the design of \emph{convertible codes}, which enable efficient transformations between codes of different parameters. In this work, we study the setting in which $\lambda$ codewords of an initial $[n^I = k^I + r^I,\, k^I]$ MDS code are merged into a single codeword of a final $[n^F = \lambda k^I + r^F,\, k^F = \lambda k^I]$ MDS code. We begin by presenting three constructions that achieve optimal \emph{access cost}, defined as the total number of disks accessed during the conversion process. The first two constructions apply when $\lambda \leq r^I$ and impose specific divisibility conditions on $r^I$ and the field size $q$. These schemes minimize both the per-symbol and the overall access cost. The third construction, which builds on a prior scheme by Kong, achieves minimal access cost while supporting arbitrary parameter regimes. All three constructions require field sizes that are linear in the final code length, and notably, the third construction achieves a field size that matches the lower bound implied by the MDS conjecture in almost all cases. In addition, we propose a construction that optimizes the \emph{bandwidth cost}, defined as the total number of symbols transmitted during conversion. This scheme is a refinement of Maturana and Rashmi's bandwidth-optimal construction based on the piggybacking framework, and achieves reduced sub-packetization.

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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

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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arXiv2018

Erasure Coding for Distributed Storage: An Overview

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 the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted. Coding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade.

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arXiv2018

Determining the Generalized Hamming Weight Hierarchy of the Binary Projective Reed-Muller Code

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 $ν$, the smallest size of the support of a subcode of ${\cal C}$ of dimension $ν$. The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work by the same authors, a code-shortening algorithm was employed to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM) codes. In the present paper, we derive a matching lower bound by adapting the proof techniques used originally for Reed-Muller (RM) codes by Wei. This results in a characterization of the GHW hierarchy of binary PRM codes.

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arXiv2017

Binary, Shortened Projective Reed Muller Codes for Coded Private Information Retrieval

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 $(n,k)$ $τ$-server binary, linear PIR code having parameters $n = \sum\limits_{i = 0}^{\ell} {m \choose i}$, $k = {m \choose \ell}$ and $τ= 2^{\ell}$ is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary, Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any $τ$ of the form $2^{\ell}$, $2^{\ell}-1$ and any value of $k$, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with other PIR codes appearing in the literature. For the particular case of $τ=3,4$, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length $n(k, τ)$ of a systematic PIR code. It follows from a result by Vardy and Yaakobi, that these codes also yield optimal, systematic primitive multi-set $(n, k, τ)_B$ batch codes for $τ=3,4$. The PIR code constructions presented here also yield upper bounds on the generalized Hamming weights of binary PRM codes.

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arXiv2016

An Explicit, Coupled-Layer Construction of a High-Rate MSR Code with Low Sub-Packetization Level, Small Field Size and All-Node Repair

This paper presents an explicit construction for an $((n,k,d=n-1), (α,β))$ 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 $n$ and where all code symbols can be repaired with the same minimum data download. The construction modifies a prior construction by Sasidharan et. al. which required far larger field-size. A building block appearing in the construction is a scalar MDS code of block length $n$. The code has a simple layered structure with coupling across layers, that allows both node repair and data recovery to be carried out by making multiple calls to a decoder for the scalar MDS code. While this work was carried out independently, there is considerable overlap with a prior construction by Ye and Barg. It is shown here that essentially the same architecture can be employed to construct MSR codes using vector binary MDS codes as building blocks in place of scalar MDS codes. The advantage here is that computations can now be carried out over a field of smaller size potentially even over the binary field as we demonstrate in an example. Further, we show how the construction can be extended to handle the case of $d<(n-1)$ under a mild restriction on the choice of helper nodes.

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arXiv2016

Outer Bounds on the Storage-Repair Bandwidth Tradeoff of Exact-Repair Regenerating Codes

In this paper, three outer bounds on the normalized storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set $\{(n,k,d),(α,β)\}$ 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 improved layered codes}, it characterizes the normalized ER tradeoff for the case $(n,k=3,d=n-1)$. It establishes a non-vanishing gap between the ER and functional-repair (FR) tradeoffs for every $(n,k,d)$. The second bound is an improvement upon an existing bound due to Mohajer et al. and is tighter than the first bound, in a regime away from the Minimum Storage Regeneraing (MSR) point. The third bound is for the case of $k=d$, under the linear setting. This outer bound matches with the achievable region of {\em layered codes} thereby characterizing the normalized ER tradeoff of linear ER codes when $k=d=n-1$.

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