Error-Correcting Codes on Projective Bundles over Deligne-Lusztig varieties

TL;DR

Problem: construct algebraic geometric codes from higher-dimensional Deligne--Lusztig varieties. Method: apply intersection theory to projective bundles over standard Deligne--Lusztig surfaces to obtain lower bounds on minimum distance and compute dimensions via $k = h^{0}(S, \mathrm{Symm}^{b}(V)\otimes\mathcal{O}_{S}(aD_{j}))$, with length $n = \#S(\mathbb{F}_{q^{\delta}})\#\mathbb{P}^{1}(\mathbb{F}_{q^{\delta}})$. Results: general bounds for rank-$2$ bundles over DL surfaces of type $A_{2}$, ${}^{2}A_{3}$, ${}^{2}A_{4}$, $C_{2}$, plus corollaries delivering explicit $n,k,d$ and two binary examples. Significance: broadens AG-code construction to higher-dimensional varieties with concrete, computable parameter sets, enabling potential practical coding schemes and guiding future work on higher-rank bundles and higher-dimensional DL varieties.

Abstract

The aim of this article is to give lower bounds on the parameters of algebraic geometric error-correcting codes constructed from projective bundles over Deligne--Lusztig surfaces. The methods based on an intensive use of the intersection theory allow us to extend the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. General bounds are obtained for the case of projective bundles of rank $2$ over standard Deligne-Lusztig surfaces, and some explicit examples coming from surfaces of type $A_{2}$ and ${}^{2}A_{4}$ are given.

Theorems & Definitions (16)

• Definition 2.1
• Example 2.2
• Theorem 2.3
• Definition 3.1
• Proposition 3.2
• Corollary 3.3
• Theorem 4.1
• proof
• Lemma 4.2
• proof