zbMATH — the first resource for mathematics

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.

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
Full Text: DOI
[1] Albert, A.A., Generalized twisted fields, Pacific J. math., 11, 1-8, (1961) · Zbl 0154.27203
[2] Albert, A.A., Isotopy for generalized twisted fields, An. acad. brasil. ci., 33, 265-275, (1961) · Zbl 0154.27301
[3] Biliotti, M.; Jha, V.; Johnson, N.L., (), 1-552
[4] Bruck, R.H.; Bose, R.C., The construction of translation planes from projective spaces, J. algebra, 1, 85-102, (1964) · Zbl 0117.37402
[5] Bruen, A.A.; Hirschfeld, J.W.P., Intersections in projective space II: pencils of quadrics, European J. combin., 9, 255-270, (1988) · Zbl 0644.51006
[6] L. Carlitz, A theorem on “ordered” polynomials in a finite field, Acta Arith. 7 (1961-1962) 167-172 · Zbl 0100.02202
[7] Cordero, M.; Figueroa, R., On some new classes of semifield planes, Osaka J. math., 30, 171-178, (1993) · Zbl 0812.51002
[8] Cordero, M.; Figueroa, R., On the semifield planes of order 5^4 and dimension 2 over the kernel, Note di mat., 22, 75-81, (2003/2004) · Zbl 1093.51009
[9] Dembowski, P., Finite geometries, (1968), Springer Verlag · Zbl 0159.50001
[10] Herzer, A.; Lunardon, G., Charakterisierung \((A, B)\)-regulärer faserungen durch schließungssätze, Geom. dedicata, 6, 471-484, (1977) · Zbl 0382.51001
[11] Hiramine, Y.; Matsumoto, M.; Oyama, T., On some extension of 1 spread sets, Osaka J. math., 24, 123-137, (1987) · Zbl 0646.51006
[12] Hirschfeld, J.W.P., Finite projective spaces of three dimensions, (1985), Clarendon Press · Zbl 0574.51001
[13] Johnson, N.L., Sequences of derivable translation planes, Osaka J. math., 25, 519-530, (1988) · Zbl 0707.51002
[14] Lunardon, G., Proposizioni configurazionali in una classe di fibrazioni, Boll. unione mat. ital., (V), 13-A, 404-413, (1976) · Zbl 0345.50012
[15] Lunardon, G., Translation ovoids, combinatorics, 2002 (maratea), J. geom., 76, 1-2, 200-215, (2003) · Zbl 1042.51008
[16] Maduram, D.M., Matrix representation of translation planes, Geom. dedicata, 4, 485-492, (1975) · Zbl 0319.50030
[17] Thas, J.A., Generalized quadrangles and flocks of cones, European J. combin., 8, 4, 441-452, (1987) · Zbl 0646.51019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.