arXiv subjects

Vinayak Ramkumar

Publications and source records attributed to Vinayak Ramkumar.

At least 19 recordsLinked to original sources

arXiv2026

CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds

In this work, we study $(r,t,x)$ quantum locally recoverable codes (qLRCs) with locality $r$, $t$ recovery sets per qudit, and intersection parameter $x$. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an $(r,t,x)$-qLRC if and only if the underlying classical codes are $(r,t,x)$ classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing $(r,t,x)$-cLRCs, which yield binary $(r,t,x)$-qLRCs via the CSS construction. For CSS $(r,t,x)$-qLRCs, we derive upper bounds on the dimension and rate, minimum-distance bounds in the pure case, and a Singleton-like dimension bound in the exact case. Finally, we show that these families attain high rates and nontrivial minimum distances.

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arXiv2026

Coding Schemes for Document Exchange under Multiple Substring Edits

We study the document exchange problem under multiple substring edits. A substring edit in a string $\mathbf{x}$ occurs when a substring $\mathbf{u}$ of $\mathbf{x}$ is replaced by an arbitrary string $\mathbf{v}$. The lengths of $\mathbf{u}$ and $\mathbf{v}$ are bounded from above by a fixed constant. Let $\mathbf{x}$ and $\mathbf{y}$ be two binary strings that differ by multiple substring edits. The aim of document exchange schemes is to construct an encoding of $\mathbf{x}$ with small length such that $\mathbf{x}$ can be recovered using $\mathbf{y}$ and the encoding. We construct a low-complexity document exchange scheme with encoding length of $4t\log n+o(\log n)$ bits, where $n$ is the length of the string $\mathbf{x}$. The best known scheme achieves an encoding length of $4t \log n+O(\log\log n)$ bits, but at a much higher computational complexity. Then, we investigate the average length of valid encodings for document exchange schemes with uniform strings $\mathbf{x}$ and develop a scheme with an expected encoding length of $(4t-1) \log n+o(\log n)$ bits. In this setting, prior works have only constructed schemes for a single substring edit.

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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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arXiv2025

Individual Confidential Computing of Polynomials over Non-Uniform Information

In this paper, we address the problem of secure distributed computation in scenarios where user data is not uniformly distributed, extending existing frameworks that assume uniformity, an assumption that is challenging to enforce in data for computation. Motivated by the pervasive reliance on single service providers for data storage and computation, we propose a privacy-preserving scheme that achieves information-theoretic security guarantees for computing polynomials over non-uniform data distributions. Our framework builds upon the concept of perfect subset privacy and employs linear hashing techniques to transform non-uniform data into approximately uniform distributions, enabling robust and secure computation. We derive leakage bounds and demonstrate that information leakage of any subset of user data to untrusted service providers, i.e., not only to colluding workers but also (and more importantly) to the admin, remains negligible under the proposed scheme.

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arXiv2024

Quantum Locally Recoverable Codes via Good Polynomials

Locally recoverable codes (LRCs) with locality parameter $r$ can recover any erased code symbol by accessing $r$ other code symbols. This local recovery property is of great interest in large-scale distributed classical data storage systems as it leads to efficient repair of failed nodes. A well-known class of optimal (classical) LRCs are subcodes of Reed-Solomon codes constructed using a special type of polynomials called good polynomials. Recently, Golowich and Guruswami initiated the study of quantum LRCs (qLRCs), which could have applications in quantum data storage systems of the future. The authors presented a qLRC construction based on good polynomials arising out of subgroups of the multiplicative group of finite fields. In this paper, we present a qLRC construction method that can employ any good polynomial. We also propose a new approach for designing good polynomials using subgroups of affine general linear groups. Golowich and Guruswami also derived a lower bound on the minimum distance of their qLRC under the restriction that $r+1$ is prime. Using similar techniques in conjunction with the expander mixing lemma, we develop minimum distance lower bounds for our qLRCs without the $r+1$ prime restriction.

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arXiv2024

$\varepsilon$-MSR Codes for Any Set of Helper Nodes

Minimum storage regenerating (MSR) codes are a class of maximum distance separable (MDS) array codes capable of repairing any single failed node by downloading the minimum amount of information from each of the helper nodes. However, MSR codes require large sub-packetization levels, which hinders their usefulness in practical settings. This led to the development of another class of MDS array codes called $\varepsilon$-MSR codes, for which the repair information downloaded from each helper node is at most a factor of $(1+\varepsilon)$ from the minimum amount for some $\varepsilon > 0$. The advantage of $\varepsilon$-MSR codes over MSR codes is their small sub-packetization levels. In previous constructions of epsilon-MSR codes, however, several specific nodes are required to participate in the repair of a failed node, which limits the performance of the code in cases where these nodes are not available. In this work, we present a construction of $\varepsilon$-MSR codes without this restriction. For a code with $n$ nodes, out of which $k$ store uncoded information, and for any number $d$ of helper nodes ($k\le d<n$), the repair of a failed node can be done by contacting any set of $d$ surviving nodes. Our construction utilizes group algebra techniques, and requires linear field size. We also generalize the construction to MDS array codes capable of repairing $h$ failed nodes using $d$ helper nodes with a slightly sub-optimal download from each helper node, for all $h \le r$ and $k \le d \le n-h$ simultaneously.

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arXiv2024

Perfect Subset Privacy in Polynomial Computation via Reed-Muller Information Super-sets

Delegating large-scale computations to service providers is a common practice which raises privacy concerns. This paper studies information-theoretic privacy-preserving delegation of data to a service provider, who may further delegate the computation to auxiliary worker nodes, in order to compute a polynomial over that data at a later point in time. We study techniques which are compatible with robust management of distributed computation systems, an area known as coded computing. Privacy in coded computing, however, has traditionally addressed the problem of colluding workers, and assumed that the server that administrates the computation is trusted. This viewpoint of privacy does not accurately reflect real-world privacy concerns, since normally, the service provider as a whole (i.e., the administrator and the worker nodes) form one cohesive entity which itself poses a privacy risk. This paper aims to shift the focus of privacy in coded computing to safeguarding the privacy of the user against the service provider as a whole, instead of merely against colluding workers inside the service provider. To this end, we leverage the recently defined notion of perfect subset privacy, which guarantees zero information leakage from all subsets of the data up to a certain size. Using known techniques from Reed-Muller decoding, we provide a scheme which enables polynomial computation with perfect subset privacy in straggler-free systems. Furthermore, by studying information super-sets in Reed-Muller codes, which may be of independent interest, we extend the previous scheme to tolerate straggling worker nodes inside the service provider.

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arXiv2024

Non-Binary Covering Codes for Low-Access Computations

Given a real dataset and a computation family, we wish to encode and store the dataset in a distributed system so that any computation from the family can be performed by accessing a small number of nodes. In this work, we focus on the families of linear computations where the coefficients are restricted to a finite set of real values. For two-valued computations, a recent work presented a scheme that gives good feasible points on the access-redundancy tradeoff. This scheme is based on binary covering codes having a certain closure property. In a follow-up work, this scheme was extended to all finite coefficient sets, using a new additive-combinatorics notion called coefficient complexity. In the present paper, we explore non-binary covering codes and develop schemes that outperform the state-of-the-art for some coefficient sets. We provide a more general coefficient complexity definition and show its applicability to the access-redundancy tradeoff.

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arXiv2023

Access-Redundancy Tradeoffs in Quantized Linear Computations

Linear real-valued computations over distributed datasets are common in many applications, most notably as part of machine learning inference. In particular, linear computations that are quantized, i.e., where the coefficients are restricted to a predetermined set of values (such as $\pm 1$), have gained increasing interest lately due to their role in efficient, robust, or private machine learning models. Given a dataset to store in a distributed system, we wish to encode it so that all such computations could be conducted by accessing a small number of servers, called the access parameter of the system. Doing so relieves the remaining servers to execute other tasks. Minimizing the access parameter gives rise to an access-redundancy tradeoff, where a smaller access parameter requires more redundancy in the system, and vice versa. In this paper, we study this tradeoff and provide several explicit low-access schemes for $\{\pm1\}$ quantized linear computations based on covering codes in a novel way. While the connection to covering codes has been observed in the past, our results strictly outperform the state-of-the-art for two-valued linear computations. We further show that the same storage scheme can be used to retrieve any linear combination with two distinct coefficients -- regardless of what those coefficients are -- with the same access parameter. This universality result is then extended to all possible quantizations with any number of values; while the storage remains identical, the access parameter increases according to a new additive-combinatorics property we call coefficient complexity. We then turn to study the coefficient complexity -- we characterize the complexity of small sets of coefficients, provide bounds, and identify coefficient sets having the highest and lowest complexity.

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arXiv2023

Private Inference in Quantized Models

A typical setup in many machine learning scenarios involves a server that holds a model and a user that possesses data, and the challenge is to perform inference while safeguarding the privacy of both parties. Private Inference has been extensively explored in recent years, mainly from a cryptographic standpoint via techniques like homomorphic encryption and multiparty computation. These approaches often come with high computational overhead and may degrade the accuracy of the model. In our work, we take a different approach inspired by the Private Information Retrieval literature. We view private inference as the task of retrieving inner products of parameter vectors with the data, a fundamental operation in many machine learning models. We introduce schemes that enable such retrieval of inner products for models with quantized (i.e., restricted to a finite set) weights; such models are extensively used in practice due to a wide range of benefits. In addition, our schemes uncover a fundamental tradeoff between user and server privacy. Our information-theoretic approach is applicable to a wide range of problems and robust in privacy guarantees for both the user and the server.

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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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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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