The classification of the largest caps in AG(5, 3)
Let us denote with \(S_{n, q}\) either the finite affine space \(\text{AG}(n, q)\) or the finite projective space \(\text{PG}(n, q)\). A \(k\)-cap \(K\) of \(S_{n, q}\) is a set of \(k\) points in \(S_{n,q}\) such that no three points are collinear. A \(k\)-cap of \(S_{n, q}\) is complete when it cannot be extended to a larger cap of \(S_{n, q}\). The main problem in the theory of caps is to find the maximal size of a cap in \(S_{n, q}\). In \(S_{2, q}\), there are at most \((q + 1)\)-caps if \(q\) is odd, and when \(q\) is even there are at most \((q + 2)\)-caps. In \(\text{AG}(3, q)\), \(q >2\), the maximal size of a cap is \(q^2\), and in \(\text{PG}(3, q)\), \(q > 2\), the maximal size of a cap is \(q^2 + 1\). And in \(S_{n,2}\) the maximal size of a cap is \(2^n\). In the paper under review the authors focus on the maximal size of a cap in \(\text{AG}(5, 3)\) and its relation to the \(56\)-cap in \(\text{PG}(3, q)\) (the Hill cap). In [\textit{A. Bruen, L. Haddad} and \textit{D. Wehlau}, Des. Codes Cryptography 13, 51-55 (1998; Zbl 0892.05010)] it is proved that the size of a cap in \(\text{AG}(5, 3)\) is at most \(48\). The Hill cap intersects a hyperplane of \(\text{PG}(5, 3)\) in either \(20\) or \(11\) points. Hence, defining \(\text{AG}(5, 3)\) to be \(\text{PG}(5, 3)\) less an \(11\)-hyperplane of this \(56\)-cap, we obtain that there exists a \(45\)-cap in \(\text{AG}(5, 3)\). The authors prove the following result: The maximal size of a cap in \(\text{AG}(5, 3)\) is equal to \(45\), and every \(45\)-cap in \(\text{AG}(5, 3)\) is obtained by deleting an \(11\)-hyperplane from a \(56\)-cap in \(\text{PG}(5, 3)\). Moreover, there is a unique type of \(45\)-caps in \(\text{AG}(5, 3)\).
scientific article; zbMATH DE number 1888403