Polynomials, Divided Differences, and Codes

TL;DR

This paper introduces a new class of multivariate codes based on divided differences, enabling efficient list decoding over all finite fields and arbitrary grids, thus overcoming previous characteristic restrictions.

Contribution

The authors develop a characteristic-insensitive code construction using divided differences, allowing for efficient list decoding on arbitrary finite grids over all finite fields.

Findings

Efficient list decoding algorithm for the new code construction.

Code can be viewed as a folded Reed-Muller code.

Decoding works over all finite fields and arbitrary grids.

Abstract

Multivariate multiplicity codes (Kopparty, Saraf, and Yekhanin, J. ACM 2014) are linear codes where the codewords are described by evaluations of multivariate polynomials (with a degree bound) and their derivatives up to a fixed order, on a suitably chosen affine point set. While good list decoding algorithms for multivariate multiplicity codes were known in some special cases of point sets by a reduction to univariate multiplicity codes, a general list decoding algorithm up to the distance of the code when the point set is an arbitrary finite grid, was obtained only recently (Bhandari et al., IEEE TIT 2023). This required the characteristic of the field to be zero or larger than the degree bound, and this requirement is somewhat necessary, since list decoding this code up to distance with small output list size is not possible when the characteristic is significantly smaller than the degree. In this work, we present an alternate construction, based on divided differences, that closely resembles the classical multiplicity codes but is `insensitive to the field characteristic'. We obtain an efficient algorithm that list decodes this code up to distance, for arbitrary finite grids and over all finite fields. Notably, our construction can be interpreted as a `folded Reed-Muller code', which might be of independent interest. The upshot of our result is that a good `Taylor-like expansion' can be expressed in terms of a good `derivative-like operator' (a divided difference), and this implies that the corresponding code admits good algorithmic list decoding.