Applying Coding Theory to Sparse Interpolation

Abstract

Fast interpolation algorithms are presented for sparse sums of characters on the product monoid $U^n $ with values in a field K in two special cases. In the first case, U is a finite cyclic group of order e, and K is a field that contains a root of unity of order e. Here linear codes over ${\mathbb{Z} / {e\mathbb{Z}}}$ can be used to construct sets of evaluation points that allow efficient interpolation by decoding Reed–Solomon codes. In the second case, K is the binary field $GF(2)$, and U is the multiplicative monoid of $GF(2)$. Here sparse sums of characters coincide with sparse Boolean polynomials and can be interpolated using the smallest set of evaluation points by decoding Reed–Muller codes.