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Semifield planes of order 81. (English) Zbl 1175.12003
The semifield planes of dimension $$2$$ over their left nucleus and dimension $$4$$ over their center have been completely classified by I. Cardinali, O. Polverino and 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.

##### MSC:
 12K10 Semifields 51A40 Translation planes and spreads in linear incidence geometry
##### Keywords:
finite semifields; computer enumeration
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