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Semifield planes of order \(q^4\) with kernel \(\mathbb F_{q^2}\) and center \(\mathbb F_q\). (English) Zbl 1108.51010
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.

MSC:
51E15 Finite affine and projective planes (geometric aspects)
51E23 Spreads and packing problems in finite geometry
12K10 Semifields
51A40 Translation planes and spreads in linear incidence geometry
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