Polynomials, Divided Differences, and Codes
S. Venkitesh
TL;DR
This work introduces a characteristic-insensitive, multivariate polynomial code built from a $\mathtt{Q}$-derivative (divided-difference) framework, unifying folded Reed-Muller and multiplicity-code perspectives. By defining the $\mathtt{Q}$-multiplicity code $\mathtt{Q}$-Mult$_{m,s}(A;k)$ and its folded RM analogue FRM$_{m,s}(A;k)$, the authors develop an interpolation-based list-decoding algorithm that works for arbitrary finite grids over any finite field, removing the prior need for the field characteristic to exceed degree bounds. The decoding strategy relies on a gluing technique and a multivariate Taylor-like expansion via $\mathsf{D}_{\mathtt{Q}}$, yielding an efficient decoder that outputs an affine subspace of dimension $O(m^2/\varepsilon)$ for radius $\delta-\varepsilon$, thus achieving capacity-like performance in a characteristic-insensitive setting. The results provide a natural folded RM interpretation and open avenues for locality properties in multivariate, char-agnostic coding schemes with practical decoding guarantees.
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.