Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties
Abstract
:1. Introduction
2. Deligne–Lusztig Varieties
- 1.
- The image is a normal, strict complete intersection. In fact, Lusztig shows (see pp. 444–445 in [18]) that Z equals the support of the scheme’s theoretic complete intersection , with mapping isomorphically onto the open subset of Z (see Table 2 in [17]). In the unitary and orthogonal cases, the singular locus of Z, consists of the finitely many translates of the closed subscheme . Hence,In the symplectic case, consists of the translates of the closed subscheme , and the previous formula becomes
- 2.
- For , and consequently
- 3.
- For any Coxeter element , we have
3. Error-Correcting Codes Construction
- Make the objects have the same codimension by blowing-up at the points;
- Make the objects have complementary dimensions, that is, make the points in some way into curves.
4. Some EC-Codes on Projective Bundles over Standard Deligne–Lusztig Surfaces
- 1.
- ;
- 2.
- ;
- 3.
- if , and otherwise,
- 1.
- ;
- 2.
- ;
- 3.
- ;
- 4.
- .
- 1.
- 2.
- 3.
- 1.
- ;
- 2.
- ;
- 3.
- .
- 1.
- ;
- 2.
- ;
- 3.
- .
- 1.
- ,
- 2.
- ,
- 3.
- and .
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Camazón Portela, D.; López Ramos, J.A. Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties. Mathematics 2023, 11, 3079. https://doi.org/10.3390/math11143079
Camazón Portela D, López Ramos JA. Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties. Mathematics. 2023; 11(14):3079. https://doi.org/10.3390/math11143079
Chicago/Turabian StyleCamazón Portela, Daniel, and Juan Antonio López Ramos. 2023. "Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties" Mathematics 11, no. 14: 3079. https://doi.org/10.3390/math11143079