Complete \(k\)-arcs in PG(\(n,q\)), \(q\) even
\(PG(n,q)\) denotes the \(n\)-dimensional projective space over the field \(GF(q)\). A \(k\)-arc of points in it with \(k\geq n+1\) is a set \(K\) of \(k\) points with the property that no \(n+1\) points of \(K\) lie in a hyperplane. It may be noted that the study of \(k\)-arcs in \(PG(n,q)\) is interesting for coding theory also -- the \(k\)-arcs of \(PG(n,q)\) and linear MDS codes of dimension \(n+1\) and length over \(GF(q)\) are equivalent objects. In this paper, the authors investigate the completeness of \(k\)-arcs in \(PG(n,q)\) where \(q\) is even. All the values of \(k\) are determined, for which there exists a complete \(k\)-arc in \(PG(n,q)\), \(q-2\geq n > q- \sqrt{q}/2-11/4\). This is proved by using the duality principle between \(k\)-arcs in \(PG(n,q)\) and dual \(k\)-arcs in \(PG(k-n-2,q)\), \((k\geq n+4)\). The theorems show that the classification of all complete \(k\)-arcs in \(PG(n,q)\), \(q\) even at \(q-2\geq n > q-\sqrt{q}/2-11/4\) is closely related to the classification of all (q+2)-arcs in \(PG(2,q)\).
scientific article; zbMATH DE number 94553
scientific article