Semifield planes of order 81 (Q1021292)

The semifield planes of dimension \(2\) over their left nucleus and dimension \(4\) over their center have been completely classified by \textit{I. Cardinali, O. Polverino} and \textit{R. Trombetti} [Eur. J. Comb. 27, No. 6, 940--961 (2006; Zbl 1108.51010)]. This implies that the planes of dimension \(2\) over one of their nuclei and dimension \(4\) over their center are all known. Therefore, if the semifield plane has order \(81,\) it was still open the case all nuclei are \(\text{GF}(3)\). In this paper the author gives a complete enumeration of such planes proving that some of them are not previously known. The enumeration has been done with a computer and the comprehensive files in GAP-format can by obtained from the homepage of the author.