Constructions of small complete arcs with prescribed symmetry

Petr Lisoněk, Stefano Marcugini, Fernanda Pambianco

Abstract


We use arcs found by Storme and van Maldeghem

in their classification of primitive arcs in ${\rm PG}(2,q)$

as seeds for constructing small complete arcs in these planes.

Our complete arcs are obtained by taking the union of

such a ``seed arc'' with some orbits of a subgroup of its stabilizer.

Using this approach we construct

five different complete 15-arcs fixed by $\Z_3$ in ${\rm PG}(2,37)$,

a complete 20-arc fixed by $\S_3$ in ${\rm PG}(2,61)$,

and two different complete 22-arcs fixed by $\D_5$ in ${\rm PG}(2,71)$.

In all three cases, the size of complete arcs constructed

in this paper is strictly smaller than the size of the smallest

complete arcs (in the respective plane) known so far.

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PID: http://hdl.handle.net/10515/sy51r6nf4

Contributions to Discrete Mathematics. ISSN: 1715-0868