On complete caps in the projective geometries over \(\mathbb F_3\)
Let \(\text{PG}(n,q)\) be the \(n\)-dimensional projective space over \(\text{GF}(q)\). A \(K\)-cap in \(\text{PG}(n,q)\) is a set of \(K\) points such that no three of them are collinear. A \(K\)-cap is said to be complete if it is maximal with respect to set-theoretic inclusion. The largest size of a cap in \(\text{PG}(5,3)\) is \(56\), but very little is known about smaller caps. In the paper under review, using both geometric and coding theoretic techniques, the authors construct complete \(48\)-caps and show that any \(53\)-cap is extendable to a \(56\)-cap. From the construction they derive new upper bounds on the largest size of a cap in \(\text{PG}(r,3)\), \(r\geq 6\).
scientific article; zbMATH DE number 1443430