Caps of order 3\(q^{2}\) in affine 4-space in characteristic 2 (Q1827584)

A cap in projective geometry PG\((n,q)\) is a set of points no three of which are collinear. The caps of PG\((n,q)\) are closely linked to codes [\textit{R. Hill}, Discrete Math. 22, 111--137 (1978; Zbl 0391.51005)]. About 50 years ago B. Segre and his school suggested and began the investigation of maximal caps in PG\((n,q)\). This problem turns out to be very difficult even for \(n=4\). At present the best known result are families of caps of order \(2.5 q^2\) (i.e. the number of points is a polynomial in \(q\) with \(2.5 q^2\) as leading term) for \(q\) odd and of order \(2 q^2\) for \(q\) even. In the paper under review the authors improve this result and produce a family of maximal caps of order \(3 q^2\) for certain \(q\) even and determine the weight distribution of the codes generated by these caps by means of their relationship with the binary Kloosterman codes (the dual Mélas codes).