Differential Codes on Higher Dimensional Varieties Via Grothendieck's Residue Symbol
David Grant, John D. Massman,, S. Srimathy
TL;DR
This work develops a higher-dimensional analogue of differential codes on curves by using Grothendieck's residue theory on an $r$-dimensional smooth projective variety $X$ over a finite field, with a finite set of $k$-rational points $\mathcal{P}$ and a system of $k$-rational divisors $\mathcal{D}$ intersecting properly. The authors define a preliminary residue-based code $C_{\Omega}(\mathcal{D},\mathcal{P},G)$ but show duality with the functional code $C_L(\mathcal{P},G)$ fails in general when intersections are not transversal or when $\mathcal{P}$ is a subset of the intersection. To address this, they introduce $(\mathcal{D},\mathcal{P})$-rectifying functions $\theta$ and the notion of rectified differential codes $C_{\Omega}(\mathcal{D},\mathcal{P},\theta,G)$, with strictly rectifying versions $\theta^s$ that yield codes that are functional (and thus dual to functional codes) on the same point set. They prove existence of such rectifying functions locally (and hence globally) and establish key properties: strictly rectified differential codes are functional, functional codes are strictly rectified, product behavior corresponds to tensor products, and orthogonality to the dual functional code holds. The results generalize Massman’s construction on transversal intersections, align with Couvreur’s surface theory, and provide a robust framework for duality-based coding on higher-dimensional varieties, with a range of examples illustrating the necessity of rectifying data for dual containment.
Abstract
We give a new construction of linear codes over finite fields on higher dimensional varieties using Grothendieck's theory of residues. This generalizes the construction of differential codes over curves to varieties of higher dimensions.