Hierarchical filtrations of line bundles and optimal algebraic geometry codes
Rahim Rahmati-asghar
TL;DR
The paper defines hierarchical depth and filtrations of line bundles to quantify internal positivity and effectivity, proving finiteness and deriving intersection-theoretic bounds including a Hodge-index–based sharp bound in the Hodge-tight case. It then develops hierarchical algebraic geometry codes by evaluating global sections along these filtrations, producing nested code families and introducing a product score $Q=(k/n)d$ whose unique maximizer identifies an optimal middle-layer code, thereby generalizing classical Goppa codes. The framework connects line-bundle theory and surface geometry to coding theory, providing explicit optimization results for curves (e.g., genus 0 cases with $i^*\approx (n-1)/2$) and Hermitian examples, and outlining extensions to higher-dimensional evaluation codes. Overall, hierarchical depth offers a systematic, geometrically grounded method to construct AG codes with favorable asymptotics and refined parameter trade-offs.
Abstract
We introduce \emph{hierarchical depth}, a new invariant of line bundles and divisors, defined via maximal chains of effective sub-line bundles. This notion gives rise to \emph{hierarchical filtrations}, refining the structure of the Picard group and providing new insights into the geometry of algebraic surfaces. We establish fundamental properties of hierarchical depth, derive inequalities through intersection theory and the Hodge index theorem, and characterize filtrations that are Hodge-tight. Using this framework, we develop a theory of \emph{hierarchical algebraic geometry codes}, constructed from evaluation spaces along these filtrations. This approach produces nested families of codes with controlled growth of parameters and identifies an optimal intermediate code maximizing a utility function balancing rate and minimum distance. Hierarchical depth thus provides a systematic method to construct AG codes with favorable asymptotic behavior, linking geometric and coding-theoretic perspectives. Our results establish new connections between line bundle theory, surface geometry, and coding theory, and suggest applications to generalized Goppa codes and higher-dimensional evaluation codes.