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Arcs of High Divisibility and Their Applications to Coding Theory

Title data

Kurz, Sascha ; Landjev, Ivan ; Pavese, Francesco ; Rousseva, Assia:
Arcs of High Divisibility and Their Applications to Coding Theory.
2023
Event: National Mathematics Colloquium of the union of bulgarian mathematicians , 14.11.2023 , Sofia, Bulgarien.
(Conference item: Lecture series , Speech )

Abstract in another language

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.

Further data

Item Type: Conference item (Speech)
Refereed: No
Additional notes: speaker: Ivan Landjev
Keywords: (t mod q)-arcs; linear codes; extension problem; Galois geometry; quadrics; caps; quasidivisible arcs; sets of type (m,n)
Subject classification: Mathematics Subject Classification Code: 51E22 (51E21 94B05)
Institutions of the University: Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematical Economics
Result of work at the UBT: No
DDC Subjects: 000 Computer Science, information, general works > 004 Computer science
500 Science > 510 Mathematics
Date Deposited: 16 Nov 2023 07:08
Last Modified: 16 Nov 2023 07:08
URI: https://eref.uni-bayreuth.de/id/eprint/87789