Limits on the Rate of Locally Testable Affine-Invariant Codes

Ben–Sasson, Eli; Sudan, Madhu

doi:10.1007/978-3-642-22935-0_35

Cited by 19 publications

(22 citation statements)

References 34 publications

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“…We prove this in Section 4.1. This definition of F + and its analysis rely centrally on some of the structural understanding of affine-invariant linear codes derived in previous works [17,10,11,6,5,4]. Lemma 4.5 allows us to say that F + is almost as nice as F, roughly analogous to the way the set of degree d + q − 1 polynomials is almost as nice as the set of degree d polynomials.…”

Section: Overview Of Proof Of Theorem 21mentioning

confidence: 99%

“…Furthermore, as shown by Kaufman and Sudan [17] affine-invariant linear codes retain some of the "locality" properties of multivariate polynomial codes (or Reed-Muller codes), such as local testability and local decodability, that have found many applications in computational complexity. This has led to a sequence of works exploring these codes, but most of the works led to codes of smaller rate than known ones, or gave alternate understanding of known codes [10,11,6,5,4]. A recent work by Guo et al [12] however changes the picture significantly.…”

Section: Introductionmentioning

confidence: 99%

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Absolutely Sound Testing of Lifted Codes

Haramaty

Ron-Zewi

Sudan

2013

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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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 form a natural abstraction of algebraic properties such as linearity and low-degree, which have been of significant interest in theoretical computer science in the past. The study of affine-invariance is motivated in part by its relationship to property testing: Affine-invariant linear codes tend to be locally testable under fairly minimal and almost necessary conditions.Recent works by Ben-Sasson et al. (CCC 2011) and Guo et al. (ITCS 2013) have introduced a new class of affine-invariant linear codes based on an operation called "lifting". Given a base code over a t-dimensional space, its m-dimensional lift consists of all words whose restriction to every t-dimensional affine subspace is a codeword of the base code. Lifting not only captures the most familiar codes, which can be expressed as lifts of low-degree polynomials, it also yields new codes when lifting "medium-degree" polynomials whose rate is better than that of corresponding polynomial codes, and all other combinatorial qualities are no worse.In this work we show that codes derived from lifting are also testable in an "absolutely sound" way. Specifically, we consider the natural test: Pick a random affine subspace of base dimension and verify that a given word is a codeword of the base code when restricted to the chosen subspace. We show that this test accepts codewords with probability one, while rejecting words at constant distance from the code with constant probability (depending only on the alphabet size). This work thus extends the results of Bhattacharyya et al. (FOCS 2010) and Haramaty et al. (FOCS 2011), while giving concrete new codes of higher rate that have absolutely sound testers.

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Section: Overview Of Proof Of Theorem 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Absolutely Sound Testing of Lifted Codes

Haramaty

Ron-Zewi

Sudan

2013

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…In sharp contrast, this question is far from being well understood in the case of testing properties of Boolean functions. In an attempt to remedy this, Sudan and several coauthors [KS08,GKS08,GKS09,BS09] have recently begun to investigate the role of invariance in property testing. The idea is that in order to be able to test if a combinatorial structure satisfies a property using very few queries to its representation, the property we are trying to test must be closed under certain transformations.…”

Section: Invariance In Testing Boolean Functionsmentioning

confidence: 99%

A Unified Framework for Testing Linear-Invariant Properties

Bhattacharyya¹,

Grigorescu²,

Shapira

2010

2010 IEEE 51st Annual Symposium on Foundations of Computer Science

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In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, Reed-Muller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linear-invariant properties, drawing on techniques from additive combinatorics and from graph theory.Our main contributions here are the following:1. We introduce a simple combinatorial condition, which we call subspace-heredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area.2. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with one-sided error, thus addressing a challenge posed by Sudan recently.3. We introduce a new technique for proving the testability of Boolean functions. Using it, we verify a special case of the conjecture.Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.

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“…This is without loss of generality, as (F Q ) t is isomorphic to F Q t for all t and prime powers Q, and this preserves affine-invariance ( [BSS11]). testability is preserved.…”

Section: Affine-invariant Codesmentioning

confidence: 99%

“…The rich structure of affine-invariance gives us some handle for understanding the constraints imposed by local testability. For example, although we know virtually no lower bounds for LTCs in the constant-query regime, it was shown in [BSS11] that affine-invariant LTCs for a constant number of queries cannot have constant rate.…”

mentioning

confidence: 99%

Limitations on Testable Affine-Invariant Codes in the High-Rate Regime

Guruswami

Sudan

Velingker

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

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Locally testable codes (LTCs) of constant minimum (absolute) distance that allow the tester to make a nearly linear number of queries have become the focus of attention recently, due to their connections to central questions in approximability theory. In particular, the binary Reed-Muller code of block length N and absolute distance d is known to be testable with O(N/d) queries, and has a dimension of ≈ N − (log N ) log d . The polylogarithmically small co-dimension is the basis of constructions of small set expanders with many "bad" eigenvalues, and size-efficient PCPs based on a shorter version of the long code. The smallest possible codimension for a distance d code (without any testability requirement) is ≈ d 2 log N , achieved by BCH codes. This raises the natural question of understanding where in the spectrum between the two classical families, Reed-Muller and BCH, the optimal co-dimension of a distance d LTC lies -in other words the "price" one has to pay for local testability.One promising approach for constructing LTCs is to focus on affine-invariant codes, whose structure makes testing guarantees easier to deduce than for general codes. Along these lines, the authors of [HRZS13] and [GKS13] recently constructed an affine-invariant family of high-rate LTCs with slightly smaller co-dimension than Reed-Muller codes. In this work, we show that their construction is essentially optimal among linear affine-invariant LTCs that contain the Reed-Muller code of the appropriate degree.

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Limits on the Rate of Locally Testable Affine-Invariant Codes

Cited by 19 publications

References 34 publications

Absolutely Sound Testing of Lifted Codes

Absolutely Sound Testing of Lifted Codes

A Unified Framework for Testing Linear-Invariant Properties

Limitations on Testable Affine-Invariant Codes in the High-Rate Regime

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