8,521 research outputs found

    High Dimensional Expansion Implies Amplified Local Testability

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    In this work, we define a notion of local testability of codes that is strictly stronger than the basic one (studied e.g., by recent works on high rate LTCs), and we term it amplified local testability. Amplified local testability is a notion close to the result of optimal testing for Reed-Muller codes achieved by Bhattacharyya et al. We present a scheme to get amplified locally testable codes from high dimensional expanders. We show that single orbit Affine invariant codes, and in particular Reed-Muller codes, can be described via our scheme, and hence are amplified locally testable. This gives the strongest currently known testability result of single orbit affine invariant codes, strengthening the celebrated result of Kaufman and Sudan

    High Dimensional Expansion Implies Amplified Local Testability

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    In this work, we define a notion of local testability of codes that is strictly stronger than the basic one (studied e.g., by recent works on high rate LTCs), and we term it amplified local testability. Amplified local testability is a notion close to the result of optimal testing for Reed-Muller codes achieved by Bhattacharyya et al. We present a scheme to get amplified locally testable codes from high dimensional expanders. We show that single orbit Affine invariant codes, and in particular Reed-Muller codes, can be described via our scheme, and hence are amplified locally testable. This gives the strongest currently known testability result of single orbit affine invariant codes, strengthening the celebrated result of Kaufman and Sudan

    Optimal Testing of Generalized Reed-Muller Codes in Fewer Queries

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    A local tester for an error correcting code CΣnC\subseteq \Sigma^{n} is a tester that makes QQ oracle queries to a given word wΣnw\in \Sigma^n and decides to accept or reject the word ww. An optimal local tester is a local tester that has the additional properties of completeness and optimal soundness. By completeness, we mean that the tester must accept with probability 11 if wCw\in C. By optimal soundness, we mean that if the tester accepts with probability at least 1ϵ1-\epsilon (where ϵ\epsilon is small), then it must be the case that ww is O(ϵ/Q)O(\epsilon/Q)-close to some codeword cCc\in C in Hamming distance. We show that Generalized Reed-Muller codes admit optimal testers with Q=(3q)d+1q1+O(1)Q = (3q)^{\lceil{ \frac{d+1}{q-1}\rceil}+O(1)} queries. Here, for a prime power q=pkq = p^{k}, the Generalized Reed-Muller code, RM[n,q,d], consists of the evaluations of all nn-variate degree dd polynomials over Fq\mathbb{F}_q. Previously, no tester achieving this query complexity was known, and the best known testers due to Haramaty, Shpilka and Sudan(which is optimal) and due to Ron-Zewi and Sudan(which was not known to be optimal) both required qd+1qq/pq^{\lceil{\frac{d+1}{q-q/p} \rceil}} queries. Our tester achieves query complexity which is polynomially better than by a power of p/(p1)p/(p-1), which is nearly the best query complexity possible for generalized Reed-Muller codes. The tester we analyze is due to Ron-Zewi and Sudan, and we show that their basic tester is in fact optimal. Our methods are more general and also allow us to prove that a wide class of testers, which follow the form of the Ron-Zewi and Sudan tester, are optimal. This result applies to testers for all affine-invariant codes (which are not necessarily generalized Reed-Muller codes).Comment: 42 pages, 8 page appendi

    Improved Local Testing for Multiplicity Codes

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    Multiplicity codes are a generalization of Reed-Muller codes which include derivatives as well as the values of low degree polynomials, evaluated in every point in ?_p^m. Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing. Recently, [Karliner et al., 2022] showed that the plane test, which tests the degree of the codeword on a random plane, is a good local tester for small enough degrees. In this work we simplify and extend the analysis of local testing for multiplicity codes, giving a more general and tight analysis. In particular, we show that multiplicity codes MRM_p(m, d, s) over prime fields with arbitrary d are locally testable by an appropriate k-flat test, which tests the degree of the codeword on a random k-dimensional affine subspace. The relationship between the degree parameter d and the required dimension k is shown to be nearly optimal, and improves on [Karliner et al., 2022] in the case of planes. Our analysis relies on a generalization of the technique of canonincal monomials introduced in [Haramaty et al., 2013]. Generalizing canonical monomials to the multiplicity case requires substantially different proofs which exploit the algebraic structure of multiplicity codes

    Enhanced Recursive Reed-Muller Erasure Decoding

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    Recent work have shown that Reed-Muller (RM) codes achieve the erasure channel capacity. However, this performance is obtained with maximum-likelihood decoding which can be costly for practical applications. In this paper, we propose an encoding/decoding scheme for Reed-Muller codes on the packet erasure channel based on Plotkin construction. We present several improvements over the generic decoding. They allow, for a light cost, to compete with maximum-likelihood decoding performance, especially on high-rate codes, while significantly outperforming it in terms of speed

    Reed-Muller codes for random erasures and errors

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    This paper studies the parameters for which Reed-Muller (RM) codes over GF(2)GF(2) can correct random erasures and random errors with high probability, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate GF(2)GF(2) polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity. The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about E(m,r)E(m,r), the matrix whose rows are truth tables of all monomials of degree r\leq r in mm variables. What is the most (resp. least) number of random columns in E(m,r)E(m,r) that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees rr, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code CC of sufficiently high rate we construct a new code CC', also of very high rate, such that for every subset SS of coordinates, if CC can recover from erasures in SS, then CC' can recover from errors in SS. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent \cite{KLP} bounds from constant degree to linear degree polynomials

    Some Applications of Coding Theory in Computational Complexity

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    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 with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs

    Low-degree tests at large distances

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    We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. In particular, we show that functions with small Gowers uniformity norms behave ``randomly'' with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem for the third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius

    Syndrome decoding of Reed-Muller codes and tensor decomposition over finite fields

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    Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk showed that for binary Reed-Muller codes of length nn and distance d=O(1)d = O(1), one can correct polylog(n)\operatorname{polylog}(n) random errors in poly(n)\operatorname{poly}(n) time (which is well beyond the worst-case error tolerance of O(1)O(1)). In this paper, we consider the problem of `syndrome decoding' Reed-Muller codes from random errors. More specifically, given the polylog(n)\operatorname{polylog}(n)-bit long syndrome vector of a codeword corrupted in polylog(n)\operatorname{polylog}(n) random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in polylog(n)\operatorname{polylog}(n) time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of Saptharishi et al. 2. The second algorithm is obtained by implementing the steps of the Berlekamp-Welch-style decoding algorithm of Saptharishi et al. in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.Comment: 24 page
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