A Fourier-Analytic Approach to Reed-Muller Decoding
We present a Fourier-analytic approach to list-decoding Reed-Muller codes over arbitrary finite fields. We use this to show that quadratic forms over any field are locally list-decodable up to their minimum distance. The analogous statement for linear polynomials was proved in the celebrated works of Goldreich et al. Previously, tight bounds for quadratic polynomials were known only for q = 2 and 3; the best bound known for other fields was the Johnson radius. Departing from previous work on Reed-Muller decoding which relies on some form of self-corrector, our work applies ideas from Fourier analysis of Boolean functions to low-degree polynomials over finite fields, in conjunction with results about the weight-distribution. We believe that the techniques used here could find other applications, we present some applications to testing and learning.
codes polynomials computational complexity fourier transforms