In $$\mathrm {PG}(2,q)$$ , the projective plane over the field $$\mathbf{F}_{q}$$ of q elements, a (k, n)-arc is a set $$\mathcal {K}$$ of k points with at most n points on any line of the plane. A fundamental question is to determine the values of k for which $$\mathcal {K}$$ is complete, that is, not contained in a $$(k+1,n)$$ -arc. In particular, what are the smallest and largest values of k for a complete $$\mathcal {K}$$ , denoted by $$t_n(2,q)$$ and $$m_n(2,q)$$ ? Here, a new lower bound for $$t_n(2,q)$$ is established and compared to known values for small q.

The known results on the maximum size of an arc in a projective space or equivalently the maximum length of a maximum distance separable linear code are surveyed. It is then shown that this maximum is q+1 for all dimensions up to q in the cases that q=11 and q=13; the result for q=11 was previously known. The strategy is to first show that a 11-arc in PG (3,11) and a 12-arc in PG (3,13) are subsets of a twisted cubic, that is, a normal rational curve.

Studies a class of projective systems and linear codes corresponding to Hermitian varieties over finite fields. The weight hierarchy, also known as the set of generalized Hamming weights, of the code is calculated. The higher weight distribution is also found.>