Abstract
We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over \({\mathbb{F}_{2}}\). We prove that, with very high probability, a random degree d + 1 polynomial has only an exponentially small correlation with all polynomials of degree d, for all degrees d up to \(\Theta\left(n\right)\). That is, a random degree d + 1 polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial. Recently, several results regarding the weight distribution of Reed–Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed–Muller codes.
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Ben-Eliezer, I., Hod, R., Lovett, S. (2009). Random Low Degree Polynomials are Hard to Approximate. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_28
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DOI: https://doi.org/10.1007/978-3-642-03685-9_28
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