Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > KEYWORD > LOCALLY TESTABLE CODES:
Reports tagged with Locally testable codes:
TR04-043 | 20th May 2004
Luca Trevisan

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs
play an important role in several recent (and old) results
in computational complexity theory. In this paper we survey
results on locally-testable and locally-decodable error-correcting
codes, and their applications to complexity theory and to
cryptography.

Locally decodable codes are error-correcting codes ... more >>>

TR04-046 | 4th June 2004

Robust Locally Testable Codes and Products of Codes

We continue the investigation of locally testable codes, i.e.,
error-correcting codes for whom membership of a given word in the
code can be tested probabilistically by examining it in very few
locations. We give two general results on local testability:
First, motivated by the recently proposed notion of robust
probabilistically ... more >>>

TR04-060 | 22nd July 2004

Simple PCPs with Poly-log Rate and Query Complexity

We give constructions of PCPs of length n*polylog(n) (with respect
to circuits of size n) that can be verified by making polylog(n)
queries to bits of the proof. These PCPs are not only shorter than
previous ones, but also simpler. Our (only) building blocks are
Reed-Solomon codes and the bivariate ... more >>>

TR05-014 | 30th January 2005
Oded Goldreich

Short Locally Testable Codes and Proofs (Survey)

We survey known results regarding locally testable codes
and locally testable proofs (known as PCPs),
with emphasis on the length of these constructs.
Locally testability refers to approximately testing
large objects based on a very small number of probes,
each retrieving a single bit in the ... more >>>

TR05-019 | 9th February 2005
Venkatesan Guruswami, Atri Rudra

Tolerant Locally Testable Codes

An error-correcting code is said to be {\em locally testable} if it has an
efficient spot-checking procedure that can distinguish codewords
from strings that are far from every codeword, looking at very few
locations of the input in doing so. Locally testable codes (LTCs) have
generated ... more >>>

TR05-046 | 17th April 2005
Irit Dinur

The PCP theorem by gap amplification

Let C={c_1,...,c_n} be a set of constraints over a set of
variables. The {\em satisfiability-gap} of C is the smallest
fraction of unsatisfied constraints, ranging over all possible
assignments for the variables.

We prove a new combinatorial amplification lemma that doubles the
satisfiability-gap of a constraint-system, with only a linear ... more >>>

TR05-086 | 14th August 2005
Dana Moshkovitz, Ran Raz

Sub-Constant Error Low Degree Test of Almost Linear Size

Revisions: 1

Given a function f:F^m \rightarrow F over a finite
field F, a low degree tester tests its proximity to
an m-variate polynomial of total degree at most d
A providing the supposed restrictions of f to
affine subspaces of ... more >>>

TR05-104 | 16th September 2005
Don Coppersmith, Atri Rudra

On the Robust Testability of Product of Codes

Ben-Sasson and Sudan in~\cite{BS04} asked if the following test
is robust for the tensor product of a code with another code--
pick a row (or column) at random and check if the received word restricted to the picked row (or column) belongs to the corresponding code. Valiant showed that ... more >>>

TR07-060 | 11th July 2007

Sparse Random Linear Codes are Locally Decodable and Testable

We show that random sparse binary linear codes are locally testable and locally decodable (under any linear encoding) with constant queries (with probability tending to one). By sparse, we mean that the code should have only polynomially many codewords. Our results are the first to show that local decodability and ... more >>>

TR07-115 | 19th November 2007
Or Meir

Combinatorial Construction of Locally Testable Codes

An error correcting code is said to be locally testable if there is a test that checks whether a given string is a codeword, or rather far from the code, by reading only a constant number of symbols of the string. Locally Testable Codes (LTCs) were first systematically studied by ... more >>>

TR09-043 | 18th May 2009
Elena Grigorescu, Tali Kaufman, Madhu Sudan

Succinct Representation of Codes with Applications to Testing

Motivated by questions in property testing, we search for linear
error-correcting codes that have the single local orbit'' property:
i.e., they are specified by a single local
constraint and its translations under the symmetry group of the
code. We show that the dual of every sparse'' binary code
whose coordinates
more >>>

TR09-128 | 29th November 2009
Gillat Kol, Ran Raz

Locally Testable Codes Analogues to the Unique Games Conjecture Do Not Exist

The Unique Games Conjecture (UGC) is possibly the most important open problem in the research of PCPs and hardness of approximation. The conjecture is a strengthening of the PCP Theorem, predicting the existence of a special type of PCP verifiers: 2-query verifiers that only make unique tests. Moreover, the UGC ... more >>>

TR09-138 | 14th December 2009
Gillat Kol, Ran Raz

Bounds on 2-Query Locally Testable Codes with Affine Tests

We study Locally Testable Codes (LTCs) that can be tested by making two queries to the tested word using an affine test. That is, we consider LTCs over a finite field F, with codeword testers that only use tests of the form $av_i + bv_j = c$, where v is ... more >>>

TR10-004 | 6th January 2010
Eli Ben-Sasson, Michael Viderman

Low Rate Is Insufficient for Local Testability

Revisions: 3

Locally testable codes (LTCs) are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations.
Kaufman and Sudan \cite{KS07} proved that sparse, low-bias linear codes are locally testable (in particular sparse random codes are locally testable).
Kopparty ... more >>>

TR10-051 | 26th March 2010

Invariance in Property Testing

Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For global properties'', i.e., properties that really depend somewhat on every piece of the data, one could ask ... more >>>

TR10-108 | 9th July 2010

Limits on the rate of locally testable affine-invariant codes

A linear code is said to be affine-invariant if the coordinates of the code can be viewed as a vector space and the code is invariant under an affine transformation of the coordinates. A code is said to be locally testable if proximity of a received word
to the code ... more >>>

TR10-119 | 27th July 2010
Michal Moshkovitz

Distance Estimators with Sublogarithmic Number of Queries

A distance estimator is a code together with a randomized algorithm. The algorithm approximates the distance of any word from the code by making a small number of queries to the word. One such example is the Reed-Muller code equipped with an appropriate algorithm. It has polynomial length, polylogarithmic alphabet ... more >>>

TR10-123 | 4th August 2010
Eli Ben-Sasson

Limitation on the rate of families of locally testable codes

Revisions: 1

This paper describes recent results which revolve around the question of the rate attainable by families of error correcting codes that are locally testable. Emphasis is placed on motivating the problem of proving upper bounds on the rate of these codes and a number of interesting open questions for future ... more >>>

TR10-130 | 18th August 2010
Tali Kaufman, Michael Viderman

Locally Testable vs. Locally Decodable Codes

Revisions: 1

We study the relation between locally testable and locally decodable codes.
Locally testable codes (LTCs) are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. Locally decodable codes (LDCs) allow to recover each message entry with ... more >>>

TR10-171 | 11th November 2010
Michael Viderman

A Note on high-rate Locally Testable Codes with sublinear query complexity

Inspired by recent construction of high-rate locally correctable codes with sublinear query complexity due to
Kopparty, Saraf and Yekhanin (2010) we address the similar question for locally testable codes (LTCs).

In this note we show a construction of high-rate LTCs with sublinear query complexity.
More formally, we show that for ... more >>>

TR10-199 | 14th December 2010
Eli Ben-Sasson, Ghid Maatouk, Amir Shpilka, Madhu Sudan

Symmetric LDPC codes are not necessarily locally testable

Locally testable codes, i.e., codes where membership in the code is testable with a constant number of queries, have played a central role in complexity theory. It is well known that a code must be a "low-density parity check" (LDPC) code for it to be locally testable, but few LDPC ... more >>>

TR10-200 | 14th December 2010
Eli Ben-Sasson, Michael Viderman

Towards lower bounds on locally testable codes via density arguments

The main open problem in the area of locally testable codes (LTCs) is whether there exists an asymptotically good family of LTCs and to resolve this question it suffices to consider the case of query complexity $3$. We argue that to refute the existence of such an asymptotically good family ... more >>>

TR11-070 | 1st May 2011
Eli Ben-Sasson, Michael Viderman

Composition of semi-LTCs by two-wise Tensor Products

In this paper we obtain a composition theorem that allows us to construct locally testable codes (LTCs) by repeated two-wise tensor products. This is the First composition theorem showing that repeating the two-wise tensor operation any constant number of times still results in a locally testable code, improving upon previous ... more >>>

TR11-087 | 3rd June 2011
Michael Viderman

A Combination of Testability and Decodability by Tensor Products

Revisions: 3

Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a very large distance are locally testable. Due to the requirement of a very large distance the associated tensor products could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result (as ... more >>>

TR12-049 | 27th April 2012
Eli Ben-Sasson, Noga Ron-Zewi, Madhu Sudan

Sparse affine-invariant linear codes are locally testable

We show that sparse affine-invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field ${\mathbb{F}}_q$ and an extension field ${\mathbb{F}}_{q^n}$, a property is a set of functions mapping ${\mathbb{F}}_{q^n}$ to ${\mathbb{F}}_q$. The property is said to be affine-invariant if it ... more >>>

TR12-106 | 27th August 2012

New affine-invariant codes from lifting

In this work we explore error-correcting codes derived from
the lifting'' of affine-invariant'' codes.
Affine-invariant codes are simply linear codes whose coordinates
are a vector space over a field and which are invariant under
affine-transformations of the coordinate space. Lifting takes codes
defined over a vector space of small dimension ... more >>>

TR12-149 | 8th November 2012
Alan Guo, Swastik Kopparty, Madhu Sudan

New affine-invariant codes from lifting

In this work we explore error-correcting codes derived from
the lifting'' of affine-invariant'' codes.
Affine-invariant codes are simply linear codes whose coordinates
are a vector space over a field and which are invariant under
affine-transformations of the coordinate space. Lifting takes codes
defined over a vector space of small dimension ... more >>>

TR12-159 | 20th November 2012
Eli Ben-Sasson, Michael Viderman

A Combinatorial Characterization of smooth LTCs and Applications

The study of locally testable codes (LTCs) has benefited from a number of nontrivial constructions discovered in recent years. Yet we still lack a good understanding of what makes a linear error correcting code locally testable and as a result we do not know what is the rate-limit of LTCs ... more >>>

TR12-168 | 26th November 2012
Michael Viderman

An error-correcting code $C \subseteq \F^n$ is called $(q,\epsilon)$-strong locally testable code (LTC) if there exists a randomized algorithm (tester) that makes at most $q$ queries to the input word. This algorithm accepts all codewords with probability 1 and rejects all non-codewords $x\notin C$ with probability at least $\epsilon \cdot ... more >>> TR13-022 | 5th February 2013 Michael Viderman Strong LTCs with inverse poly-log rate and constant soundness Revisions: 1 An error-correcting code$C \subseteq \F^n$is called$(q,\epsilon)$-strong locally testable code (LTC) if there exists a tester that makes at most$q$queries to the input word. This tester accepts all codewords with probability 1 and rejects all non-codewords$x\notin C$with probability at least$\epsilon \cdot \delta(x,C)$, where ... more >>> TR13-030 | 20th February 2013 Elad Haramaty, Noga Ron-Zewi, Madhu Sudan Absolutely Sound Testing of Lifted Codes In this work we present a strong analysis of the testability of a broad, and to date the most interesting known, class of "affine-invariant'' codes. Affine-invariant codes are codes whose coordinates are associated with a vector space and are invariant under affine transformations of the coordinate space. Affine-invariant linear codes ... more >>> TR13-073 | 14th May 2013 Oded Goldreich On the Communication Complexity Methodology for Proving Lower Bounds on the Query Complexity of Property Testing Revisions: 2 A couple of years ago, Blais, Brody, and Matulef put forward a methodology for proving lower bounds on the query complexity of property testing via communication complexity. They provided a restricted formulation of their methodology (via simple combining operators'') and also hinted towards a more general formulation, which we spell ... more >>> TR13-114 | 24th August 2013 Parikshit Gopalan, Salil Vadhan, Yuan Zhou Locally Testable Codes and Cayley Graphs Revisions: 1 We give two new characterizations of ($\F_2$-linear) locally testable error-correcting codes in terms of Cayley graphs over$\F_2^h$: \begin{enumerate} \item A locally testable code is equivalent to a Cayley graph over$\F_2^h$whose set of generators is significantly larger than$h$and has no short linear dependencies, but yields a ... more >>> TR13-122 | 5th September 2013 Irit Dinur, Venkatesan Guruswami PCPs via low-degree long code and hardness for constrained hypergraph coloring Revisions: 1 We develop new techniques to incorporate the recently proposed short code" (a low-degree version of the long code) into the construction and analysis of PCPs in the classical Label Cover + Fourier Analysis'' framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs ... more >>> TR14-025 | 25th February 2014 Oded Goldreich, Tom Gur, Ilan Komargodski Strong Locally Testable Codes with Relaxed Local Decoders Locally testable codes (LTCs) are error-correcting codes that admit very efficient codeword tests. An LTC is said to be strong if it has a proximity-oblivious tester; that is, a tester that makes only a constant number of queries and reject non-codewords with probability that depends solely on their distance from ... more >>> TR14-029 | 4th March 2014 Oded Goldreich, Dana Ron On Learning and Testing Dynamic Environments Revisions: 3 We initiate a study of learning and testing dynamic environments, focusing on environment that evolve according to a fixed local rule. The (proper) learning task consists of obtaining the initial configuration of the environment, whereas for non-proper learning it suffices to predict its future values. The testing task consists of ... more >>> TR14-067 | 4th May 2014 Venkatesan Guruswami, Madhu Sudan, Ameya Velingker, Carol Wang Limitations on Testable Affine-Invariant Codes in the High-Rate Regime Locally testable codes (LTCs) of constant distance that allow the tester to make a linear number of queries have become the focus of attention recently, due to their elegant connections to hardness of approximation. In particular, the binary Reed-Muller code of block length$N$and distance$d$is known to ... more >>> TR15-020 | 31st January 2015 Michael Viderman Explicit Strong LTCs with inverse poly-log rate and constant soundness Revisions: 1 An error-correcting code$C \subseteq \F^n$is called$(q,\epsilon)$-strong locally testable code (LTC) if there exists a tester that makes at most$q$queries to the input word. This tester accepts all codewords with probability 1 and rejects all non-codewords$x\notin C$with probability at least$\epsilon \cdot \delta(x,C)$, where ... more >>> TR15-043 | 2nd April 2015 Alan Guo, Elad Haramaty, Madhu Sudan Robust testing of lifted codes with applications to low-degree testing A local tester for a code probabilistically looks at a given word at a small set of coordinates and based on this local view accepts codewords with probability one while rejecting words far from the code with constant probabilility. A local tester for a code is said to be robust'' ... more >>> TR15-068 | 21st April 2015 Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf High rate locally-correctable and locally-testable codes with sub-polynomial query complexity Revisions: 2 In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length$n$, constant rate (which can even be taken arbitrarily close to 1), constant ... more >>> TR15-110 | 8th July 2015 Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity Revisions: 1 An error correcting code is said to be \emph{locally testable} if there is a test that checks whether a given string is a codeword, or rather far from the code, by reading only a small number of symbols of the string. Locally testable codes (LTCs) are both interesting in their ... more >>> TR16-042 | 19th March 2016 Oded Goldreich, Tom Gur Universal Locally Testable Codes Revisions: 2 We initiate a study of universal locally testable codes" (universal-LTCs). These codes admit local tests for membership in numerous possible subcodes, allowing for testing properties of the encoded message. More precisely, a universal-LTC$C:\{0,1\}^k \to \{0,1\}^n$for a family of functions$\mathcal{F} = \{ f_i : \{0,1\}^k \to \{0,1\} \}_{i ... more >>>

TR16-125 | 31st July 2016
Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, Elena Grigorescu

Local Testing for Membership in Lattices

Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in ... more >>>

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

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

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 >>>

TR17-110 | 22nd June 2017
Alessandro Chiesa, Peter Manohar, Igor Shinkar

On Axis-Parallel Tests for Tensor Product Codes

Many low-degree tests examine the input function via its restrictions to random hyperplanes of a certain dimension. Examples include the line-vs-line (Arora, Sudan 2003), plane-vs-plane (Raz, Safra 1997), and cube-vs-cube (Bhangale, Dinur, Livni 2017) tests.

In this paper we study a test introduced by Ben-Sasson and Sudan in 2006 that ... 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 >>>

ISSN 1433-8092 | Imprint