Constructions of small complete arcs with prescribed symmetry

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

The geometry of the plane of order nineteen and its application to error-correcting codes

In the projective space PG(k−1; q) over Fq, the finite field of order q, an (n; r)-arc K is a set of n points with at most r on a hyperplane and there is some hyperplane meeting K in exactly r points. An arc is complete if it is maximal with respect to inclusion. The arc K corresponds to a projective [n; k;n − r]q-code of length n, dimension k, and minimum distance n − r; if K is a complete arc, then the corresponding projective code cannot be extended. In this thesis, the n-sets in PG(1; 19) up to n = 10 and the n-arcs in PG(2; 19) for 4 B n B 20 in both the complete and incomplete cases are classified. The set of rational points of a non-singular, plane cubic curve can be considered as an arc of degree three. Over F19, these curves are classified, and the maximum size of the complete arc of degree three that can be constructed from each such incomplete arc is given

Primitive arcs in PG(2,Q)

AbstractWe show that a complete arc K in the projective plane PG(2, q) admitting a transitive primitive group of projective transformations is either a cyclic arc of prime order or a known arc. If the completeness assumption is dropped, then K has either an affine primitive group, or K is contained in an explicit list. In order to find these primitive arcs, it is necessary to determine all complete k-arcs fixed by a projective elementary abelian group of order k. As a corollary to our result, we list all complete arcs fixed by a 2-transitive projective group