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Kurz, Sascha ; Landjev, Ivan ; Pavese, Francesco ; Rousseva, Assia:
Arcs of High Divisibility and Their Applications to Coding Theory.
2023
Veranstaltung: National Mathematics Colloquium of the union of bulgarian mathematicians
, 14.11.2023
, Sofia, Bulgarien.
(Veranstaltungsbeitrag: Vortragsreihe
,
Vortrag
)
Abstract
The (t mod q)-arcs were introduced as a tool for an unified treatment of the extension problem for linear codes. An arc K in PG(r, q) is called a (t mod q)-arc if K(L) is congruent to t modulo q for every line L from PG(r, q). If in addition each point has multiplicity at most t, then K is called a strong (t mod q)-arc. There exists a general lifting construction for (strong) (t mod q)-arcs which given a (t mod q) arc in PG(r, q) produces such an arc in PG(r + 1, q). It was conjectured that all strong indecomposable (t mod q)-arcs in PG(r, q) for r at least 3 3 are lifted. This conjecture turned out to be wrong. Three exceptional (3 mod 5)-arcs in PG(3, 5) of respective sizes 128, 143 and 168 that are not lifted were constructed by computer. This result was used to fill in the gap in the non-existence proof for the putative [104, 4, 82]_5-code. A geometric (computerfree) description of the three exceptional (3 mod 5)-arcs was presented in a recent paper.
One of them uses the Abatangelo-Korchmaros-Larato cap of size 20 in PG(3, 5), while the other two are based on the elliptic and hyperbolic quadrics. In this talk, we present a geometric description of the three exceptional (3 mod 5)-arcs in PG(3, 5) and prove that every strong (3 mod 5)-arc in PG(r, 5), r at least 4, is either lifted or a quadratic arc.
Weitere Angaben
| Publikationsform: | Veranstaltungsbeitrag (Vortrag) |
|---|---|
| Begutachteter Beitrag: | Nein |
| Zusätzliche Informationen: | speaker: Ivan Landjev |
| Keywords: | (t mod q)-arcs; linear codes; extension problem; Galois geometry; quadrics; caps; quasidivisible arcs; sets of type (m,n) |
| Fachklassifikationen: | Mathematics Subject Classification Code: 51E22 (51E21 94B05) |
| Institutionen der Universität: | Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Wirtschaftsmathematik |
| Titel an der UBT entstanden: | Nein |
| Themengebiete aus DDC: | 000 Informatik,Informationswissenschaft, allgemeine Werke > 004 Informatik 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Eingestellt am: | 16 Nov 2023 07:08 |
| Letzte Änderung: | 16 Nov 2023 07:08 |
| URI: | https://eref.uni-bayreuth.de/id/eprint/87789 |