an:05030993
Zbl 1108.51010
Cardinali, I.; Polverino, O.; Trombetti, R.
Semifield planes of order \(q^4\) with kernel \(\mathbb F_{q^2}\) and center \(\mathbb F_q\)
EN
Eur. J. Comb. 27, No. 6, 940-961 (2006).
0195-6698
2006
j
51E15 51E23 12K10 51A40
semifield plane; semifield spread; hyperbolic quadric
All semifield planes of order \(q^4\) with kernel of order \(q^2\) and center of order \(q\) are classified. Indeed the authors give a complete list up to isomorphy (i.e., up to isotopy of the underlying semifields). A semifield plane can be described by a spread \(S\). In the present situation, \(S\) is a set of non-singular \(2\times2\)-matrices over \(\text{GF}(q^2)\), which form a \(\text{GF}(q)\) vector space. Viewed in the projective space \(\text{PG}(3,q^2)\) the set \(S\) defines a (\(\text{GF}(q)\)-linear) subset disjoint from the quadric defined by the determinant. This equivalent description is actually used to carry out the classification. In particular, it is shown that projective equivalence of such subsets in \(\text{PG}(3,q^2)\) is the same as isomorphy of the corresponding planes.
Hubert Kiechle (Hamburg)