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Electronic Colloquium on Computational Complexity

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REPORTS > KEYWORD > CODING THEORY:
Reports tagged with Coding theory:
TR06-128 | 5th October 2006
Shankar Kalyanaraman, Chris Umans

On obtaining pseudorandomness from error-correcting codes.

A number of recent results have constructed randomness extractors
and pseudorandom generators (PRGs) directly from certain
error-correcting codes. The underlying construction in these
results amounts to picking a random index into the codeword and
outputting $m$ consecutive symbols (the codeword is obtained from
the weak random source in the case ... more >>>


TR07-073 | 3rd August 2007
Parikshit Gopalan, Subhash Khot, Rishi Saket

Hardness of Reconstructing Multivariate Polynomials over Finite Fields

We study the polynomial reconstruction problem for low-degree
multivariate polynomials over finite fields. In the GF[2] version of this problem, we are given a set of points on the hypercube and target values $f(x)$ for each of these points, with the promise that there is a polynomial over GF[2] of ... more >>>


TR07-089 | 13th September 2007
Parikshit Gopalan, Venkatesan Guruswami

Deterministic Hardness Amplification via Local GMD Decoding

We study the average-case hardness of the class NP against
deterministic polynomial time algorithms. We prove that there exists
some constant $\mu > 0$ such that if there is some language in NP
for which no deterministic polynomial time algorithm can decide L
correctly on a $1- (log n)^{-\mu}$ fraction ... more >>>


TR07-098 | 2nd October 2007
Tali Kaufman, Simon Litsyn, Ning Xie

Breaking the $\epsilon$-Soundness Bound of the Linearity Test over GF(2)

For Boolean functions that are $\epsilon$-far from the set of linear functions, we study the lower bound on the rejection probability (denoted $\textsc{rej}(\epsilon)$) of the linearity test suggested by Blum, Luby and Rubinfeld. The interest in this problem is partly due to its relation to PCP constructions and hardness of ... more >>>


TR11-064 | 23rd April 2011
Mark Braverman

Towards deterministic tree code constructions

We present a deterministic operator on tree codes -- we call tree code product -- that allows one to deterministically combine two tree codes into a larger tree code. Moreover, if the original tree codes are efficiently encodable and decodable, then so is their product. This allows us to give ... more >>>


TR13-046 | 27th March 2013
Venkatesan Guruswami, Chaoping Xing

Optimal rate list decoding of folded algebraic-geometric codes over constant-sized alphabets

We construct a new list-decodable family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. The function fields underlying these codes are constructed using class field theory, specifically Drinfeld modules of rank $1$, and designed to have an automorphism of large order that is used to ``fold" the AG code. ... more >>>


TR13-118 | 2nd September 2013
Mahdi Cheraghchi, Venkatesan Guruswami

Capacity of Non-Malleable Codes

Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), encode messages $s$ in a manner so that tampering the codeword causes the decoder to either output $s$ or a message that is independent of $s$. While this is an impossible goal to achieve against unrestricted tampering functions, rather surprisingly ... more >>>


TR13-121 | 4th September 2013
Mahdi Cheraghchi, Venkatesan Guruswami

Non-Malleable Coding Against Bit-wise and Split-State Tampering

Revisions: 1

Non-malleable coding, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error-detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable ... more >>>


TR14-087 | 12th July 2014
Abhishek Bhowmick, Shachar Lovett

List decoding Reed-Muller codes over small fields

Revisions: 1

The list decoding problem for a code asks for the maximal radius up to which any ball of that radius contains only a constant number of codewords. The list decoding radius is not well understood even for well studied codes, like Reed-Solomon or Reed-Muller codes.

Fix a finite field $\mathbb{F}$. ... more >>>


TR14-127 | 11th October 2014
Alexandros G. Dimakis, Anna Gal, Ankit Singh Rawat, Zhao Song

Batch Codes through Dense Graphs without Short Cycles

Consider a large database of $n$ data items that need to be stored using $m$ servers.
We study how to encode information so that a large number $k$ of read requests can be performed \textit{in parallel} while the rate remains constant (and ideally approaches one).
This problem is equivalent ... more >>>


TR14-165 | 3rd December 2014
Venkatesan Guruswami, Ameya Velingker

An Entropy Sumset Inequality and Polynomially Fast Convergence to Shannon Capacity Over All Alphabets

We prove a lower estimate on the increase in entropy when two copies of a conditional random variable $X | Y$, with $X$ supported on $\mathbb{Z}_q=\{0,1,\dots,q-1\}$ for prime $q$, are summed modulo $q$. Specifically, given two i.i.d. copies $(X_1,Y_1)$ and $(X_2,Y_2)$ of a pair of random variables $(X,Y)$, with $X$ ... more >>>


TR15-014 | 18th January 2015
Noga Alon, Mark Braverman, Klim Efremenko, Ran Gelles, Bernhard Haeupler

Reliable Communication over Highly Connected Noisy Networks

We consider the task of multiparty computation performed over networks in
the presence of random noise. Given an $n$-party protocol that takes $R$
rounds assuming noiseless communication, the goal is to find a coding
scheme that takes $R'$ rounds and computes the same function with high
probability even when the ... more >>>


TR16-040 | 16th March 2016
Baris Aydinlioglu, Eric Bach

Affine Relativization: Unifying the Algebrization and Relativization Barriers

Revisions: 3

We strengthen existing evidence for the so-called "algebrization barrier". Algebrization --- short for algebraic relativization --- was introduced by Aaronson and Wigderson (AW) in order to characterize proofs involving arithmetization, simulation, and other "current techniques". However, unlike relativization, eligible statements under this notion do not seem to have basic closure ... more >>>


TR16-090 | 27th May 2016
Bernhard Haeupler, Ameya Velingker

Bridging the Capacity Gap Between Interactive and One-Way Communication

We study the communication rate of coding schemes for interactive communication that transform any two-party interactive protocol into a protocol that is robust to noise.

Recently, Haeupler (FOCS '14) showed that if an $\epsilon > 0$ fraction of transmissions are corrupted, adversarially or randomly, then it is possible to ... more >>>


TR16-166 | 1st November 2016
Mark Braverman, Ran Gelles, Michael A. Yitayew

Optimal Resilience for Short-Circuit Noise in Formulas

Revisions: 1

We show an efficient method for converting a logic circuit of gates with fan-out 1 into an equivalent circuit that works even if some fraction of its gates are short-circuited, i.e., their output is short-circuited to one of their inputs. Our conversion can be applied to any circuit with fan-in ... more >>>


TR16-192 | 25th November 2016
Oded Goldreich, Tom Gur

Universal Locally Verifiable Codes and 3-Round Interactive Proofs of Proximity for CSP

Revisions: 2 , Comments: 1

Universal locally testable codes (Universal-LTCs), recently introduced in our companion paper [GG16], are codes that admit local tests for membership in numerous possible subcodes, allowing for testing properties of the encoded message. In this work, we initiate the study of the NP analogue of these codes, wherein the testing procedures ... more >>>


TR17-064 | 20th April 2017
Venkatesan Guruswami, Chaoping Xing, chen yuan

Subspace Designs based on Algebraic Function Fields

Subspace designs are a (large) collection of high-dimensional subspaces $\{H_i\}$ of $\F_q^m$ such that for any low-dimensional subspace $W$, only a small number of subspaces from the collection have non-trivial intersection with $W$; more precisely, the sum of dimensions of $W \cap H_i$ is at most some parameter $L$. The ... more >>>


TR17-126 | 7th August 2017
Swastik Kopparty, Shubhangi Saraf

Local Testing and Decoding of High-Rate Error-Correcting Codes

We survey the state of the art in constructions of locally testable
codes, locally decodable codes and locally correctable codes of high rate.

more >>>

TR17-143 | 26th September 2017
Tom Gur, Govind Ramnarayan, Ron Rothblum

Relaxed Locally Correctable Codes

Revisions: 1

Locally decodable codes (LDCs) and locally correctable codes (LCCs) are error-correcting codes in which individual bits of the message and codeword, respectively, can be recovered by querying only few bits from a noisy codeword. These codes have found numerous applications both in theory and in practice.

A natural relaxation of ... more >>>


TR18-008 | 10th January 2018
Tom Gur, Igor Shinkar

An Entropy Lower Bound for Non-Malleable Extractors

A (k,\eps)-non-malleable extractor is a function nmExt : {0,1}^n x {0,1}^d -> {0,1} that takes two inputs, a weak source X~{0,1}^n of min-entropy k and an independent uniform seed s in {0,1}^d, and outputs a bit nmExt(X, s) that is \eps-close to uniform, even given the seed s and the ... more >>>


TR18-017 | 26th January 2018
Venkatesan Guruswami, Nicolas Resch, Chaoping Xing

Lossless dimension expanders via linearized polynomials and subspace designs

For a vector space $\mathbb{F}^n$ over a field $\mathbb{F}$, an $(\eta,\beta)$-dimension expander of degree $d$ is a collection of $d$ linear maps $\Gamma_j : \mathbb{F}^n \to \mathbb{F}^n$ such that for every subspace $U$ of $\mathbb{F}^n$ of dimension at most $\eta n$, the image of $U$ under all the maps, $\sum_{j=1}^d ... more >>>


TR18-096 | 13th May 2018
Venkatesan Guruswami, Andrii Riazanov

Beating Fredman-Komlós for perfect $k$-hashing

We say a subset $C \subseteq \{1,2,\dots,k\}^n$ is a $k$-hash code (also called $k$-separated) if for every subset of $k$ codewords from $C$, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as $(\log_2 |C|)/n$, of a $k$-hash code is ... more >>>


TR19-005 | 16th January 2019
Omar Alrabiah, Venkatesan Guruswami

An Exponential Lower Bound on the Sub-Packetization of MSR Codes

An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a ... more >>>


TR19-056 | 11th April 2019
Tom Gur, Oded Lachish

A Lower Bound for Relaxed Locally Decodable Codes

Revisions: 1

A locally decodable code (LDC) C:{0,1}^k -> {0,1}^n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ranging from probabilistically checkable proofs to ... more >>>


TR20-047 | 16th April 2020
Ronen Shaltiel, Jad Silbak

Explicit Uniquely Decodable Codes for Space Bounded Channels That Achieve List-Decoding Capacity

Revisions: 2

We consider codes for space bounded channels. This is a model for communication under noise that was introduced by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in ... more >>>


TR20-113 | 27th July 2020
Alessandro Chiesa, Tom Gur, Igor Shinkar

Relaxed Locally Correctable Codes with Nearly-Linear Block Length and Constant Query Complexity

Locally correctable codes (LCCs) are error correcting codes C : \Sigma^k \to \Sigma^n which admit local algorithms that correct any individual symbol of a corrupted codeword via a minuscule number of queries. This notion is stronger than that of locally decodable codes (LDCs), where the goal is to only recover ... more >>>


TR20-149 | 29th September 2020
Oded Goldreich, Avi Wigderson

Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing

Revisions: 2


A graph $G$ is called {\em self-ordered}\/ (a.k.a asymmetric) if the identity permutation is its only automorphism.
Equivalently, there is a unique isomorphism from $G$ to any graph that is isomorphic to $G$.
We say that $G=(V,E)$ is {\em robustly self-ordered}\/ if the size of the symmetric difference ... more >>>


TR20-154 | 10th October 2020
Marcel Dall'Agnol, Tom Gur, Oded Lachish

A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Privacy

We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes $q$ adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with $n^{1- 1/O(q^2 ... more >>>


TR20-156 | 22nd October 2020
Sankeerth Rao Karingula, Shachar Lovett

Codes over integers, and the singularity of random matrices with large entries

Revisions: 1

The prototypical construction of error correcting codes is based on linear codes over finite fields. In this work, we make first steps in the study of codes defined over integers. We focus on Maximum Distance Separable (MDS) codes, and show that MDS codes with linear rate and distance can be ... more >>>


TR20-162 | 7th November 2020
Dean Doron, Mary Wootters

High-Probability List-Recovery, and Applications to Heavy Hitters

Revisions: 1

An error correcting code $\mathcal{C} \colon \Sigma^k \to \Sigma^n$ is list-recoverable from input list size $\ell$ if for any sets $\mathcal{L}_1, \ldots, \mathcal{L}_n \subseteq \Sigma$ of size at most $\ell$, one can efficiently recover the list $\mathcal{L} = \{ x \in \Sigma^k : \forall j \in [n], \mathcal{C}(x)_j \in \mathcal{L}_j ... more >>>


TR20-192 | 27th December 2020
Oded Goldreich, Avi Wigderson

Constructing Large Families of Pairwise Far Permutations: Good Permutation Codes Based on the Shuffle-Exchange Network


We consider the problem of efficiently constructing an as large as possible family of permutations such that each pair of permutations are far part (i.e., disagree on a constant fraction of their inputs).
Specifically, for every $n\in\N$, we present a collection of $N=N(n)=(n!)^{\Omega(1)}$ pairwise far apart permutations $\{\pi_i:[n]\to[n]\}_{i\in[N]}$ and ... more >>>


TR21-036 | 14th March 2021
Siddharth Bhandari, Prahladh Harsha, Mrinal Kumar, Madhu Sudan

Ideal-theoretic Explanation of Capacity-achieving Decoding

In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their ... more >>>




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