Encoding and decoding of Reed-Muller codes with quasi-linear complexity

Encoding and decoding of Reed-Muller codes have been a major research topic in coding and theoretical computer science communities. Despite of the fact that there have been numerous encoding and decoding algorithms in the literature, most of them are not quasi-linear time algorithms for arbitrary order Reed-Muller codes. Under the decoding framework proposed by Pellikaan and Wu (IEEE TIT, 2004) which regards Reed-Muller codes as subfield subcodes of Reed-Solomon codes, we propose a new decoding algorithm for Reed-Muller codes that improves previous polynomial decoding complexity to quasilinear complexity. Our new decoding algorithm includes multivariate multipoint evaluation (MPE) and interpolation under a new basis of the multivariate polynomial space as two main steps. We show that the MPE and interpolation at certain multipoint sets can be performed in quasi-linear time as well. Our approach is based on a well-known transform between univariate polynomials and multivariate polynomials. We make use of the key fact that the transformation matrix between univariate polynomials and multivariate polynomials is sparse. Due to sparsity, MPE and interpolation of multivariate polynomials and decoding of Reed-Muller codes can be reduced to MPE and interpolation of univariate polynomials and decoding of Reed-Solomon codes without extra cost respectively, i.e, the complexity of MPE and interpolation of multivariate polynomials (and, respectively, decoding of Reed-Muller codes) is dominated by that of MPE and interpolation of univariate polynomials (and, respectively, decoding of Reed-Solomon codes). As a result of this reduction, we obtain our quasi-linear time algorithms.