On the (4\nu,3)-arcs in PG(2,q) and the related linear codes

In this paper, we use an irreducible plane-cubic curves in the projective plane PG(2,q) to construct (k,3)-arcs of size 4\nu where \lceil \frac{q+1-2\sqrt{q}}{4}\rceil\leq\nu\leq\lfloor \frac{q+1+2\sqrt{q}}{4} \rfloor. Each of these arcs gives rise to an error-correcting code that corrects the maximum possible number of errors for its length. Furthermore, we discuss the completeness of each arc. The isotropy subgroup of each arc are determined. All Griesmer codes that correspond to plane-cubic curves are given for 7\leq q\leq 37,q is a prime.

On the (4\nu,3)-arcs in PG(2,q) and the related linear codes Keywords:

Cubic curves , Arcs , Projective plane , codes , stabilizer groups