Search for: All records

In this work, we show that the class of multivariate degree-d polynomials mapping {0,1}n to any Abelian group G is locally correctable with Õd((log n )d) queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of Amireddy, Behera, Paraashar, Srinivasan, and Sudan [1] (STOC 2024) who considered the case of linear polynomials (d = 1) and gave analogous results. Low-degree polynomials over the Boolean cube {0,1}n arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [1] from linear polynomials to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree polynomials over the Boolean cube. For local correction, we construct a set of points in the Boolean cube that lie between two exponentially close parallel hyperplanes and is moreover an interpolating set for degree-d polynomials. To show that the class of degree-d polynomials is list decodable up to the minimum distance, we stitch together results on anti-concentration of low-degree polynomials, the Sunflower lemma, and the Footprint bound for counting common zeroes of polynomials. Analyzing the local list corrector of [1] for higher degree polynomials involves understanding random restrictions of non-zero degree-d polynomials on a Hamming slice. In particular, we show that a simple random restriction process for reducing the dimension of the Boolean cube is a suitably good sampler for Hamming slices. Thus our exploration unearths several new techniques that are useful in understanding the combinatorial structure of low-degree polynomials over {0, 1}n. 

In this work, we show that the class of multivariate degree-d polynomials mapping {0,1}n to any Abelian group G is locally correctable with Õd((log n )d) queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of Amireddy, Behera, Paraashar, Srinivasan, and Sudan [1] (STOC 2024) who considered the case of linear polynomials (d = 1) and gave analogous results. Low-degree polynomials over the Boolean cube {0,1}n arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [1] from linear polynomials to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree polynomials over the Boolean cube. For local correction, we construct a set of points in the Boolean cube that lie between two exponentially close parallel hyperplanes and is moreover an interpolating set for degree-d polynomials. To show that the class of degree-d polynomials is list decodable up to the minimum distance, we stitch together results on anti-concentration of low-degree polynomials, the Sunflower lemma, and the Footprint bound for counting common zeroes of polynomials. Analyzing the local list corrector of [1] for higher degree polynomials involves understanding random restrictions of non-zero degree-d polynomials on a Hamming slice. In particular, we show that a simple random restriction process for reducing the dimension of the Boolean cube is a suitably good sampler for Hamming slices. Thus our exploration unearths several new techniques that are useful in understanding the combinatorial structure of low-degree polynomials over {0, 1}n. 

In this work, we show that the class of multivariate degree-d polynomials mapping {0,1}n to any Abelian group G is locally correctable with O˜d((logn)d) queries for up to a fraction of errors approaching half the minimum distance of the underlying code. In particular, this result holds even for polynomials over the reals or the rationals, special cases that were previously not known. Further, we show that they are locally list correctable up to a fraction of errors approaching the minimum distance of the code. These results build on and extend the prior work of the authors [ABPSS24] (STOC 2024) who considered the case of linear polynomials and gave analogous results. Low-degree polynomials over the Boolean cube {0,1}n arise naturally in Boolean circuit complexity and learning theory, and our work furthers the study of their coding-theoretic properties. Extending the results of [ABPSS24] from linear to higher-degree polynomials involves several new challenges and handling them gives us further insights into properties of low-degree polynomials over the Boolean cube. For local correction, we construct a set of points in the Boolean cube that lie between two exponentially close parallel hyperplanes and is moreover an interpolating set for degree-d polynomials. To show that the class of degree-d polynomials is list decodable up to the minimum distance, we stitch together results on anti-concentration of low-degree polynomials, the Sunflower lemma, and the Footprint bound for counting common zeroes of polynomials. Analyzing the local list corrector of [ABPSS24] for higher degree polynomials involves understanding random restrictions of non-zero degree-d polynomials on a Hamming slice. In particular, we show that a simple random restriction process for reducing the dimension of the Boolean cube is a suitably good sampler for Hamming slices. 

We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain {0,1}n over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance 1/2 and we give local-correction algorithms correcting up to nearly 1/4-fraction errors making O(logn) queries. This query complexity is optimal up to poly(loglogn) factors. We also give local list-correcting algorithms correcting (1/2 − ε)-fraction errors with Oε(logn) queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is ℤ2. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). Previous works in the area mostly focused on the case where the domain is a vector space or a group and this lends to tools that exploit symmetry. Since our domains lack such symmetries, we encounter new challenges whose resolution may be of independent interest. The central challenge in constructing the local corrector is constructing “nearly balanced vectors” over {−1,1}n that span 1n — we show how to construct O(logn) vectors that do so, with entries in each vector summing to ±1. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius (1/2−ε) is Oε(1). Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls. 

We consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain {0,1}n over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance 1/2 and we give local-correction algorithms correcting up to nearly 1/4-fraction errors making O(logn) queries. This query complexity is optimal up to poly(loglogn) factors. We also give local list-correcting algorithms correcting (1/2 − ε)-fraction errors with Oε(logn) queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is ℤ2. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). Previous works in the area mostly focused on the case where the domain is a vector space or a group and this lends to tools that exploit symmetry. Since our domains lack such symmetries, we encounter new challenges whose resolution may be of independent interest. The central challenge in constructing the local corrector is constructing “nearly balanced vectors” over {−1,1}n that span 1n — we show how to construct O(logn) vectors that do so, with entries in each vector summing to ±1. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius (1/2−ε) is Oε(1). Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.