Small complete caps in three-dimensional Galois spaces

In a three-dimensional Galois space of odd order q, the smallest complete caps appeared in the literature have size approximately qq/2 and were presented by Pellegrino in 1998. In this paper, a major gap in the proof of their completeness is pointed out. On the other hand, we show that a variant of Pellegrino's method provides the smallest known complete caps for each odd q between 100 and 30 000.