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Infinite families of new semifields. (English) Zbl 1224.12010

The authors construct six new infinite families of finite semifields, all of which are two-dimensional over their left nuclei. The semifields are given by providing spread sets of linear mappings. It is shown that semifields in different families are never isotopic. The classification of isotopy classes within a given family is still ongoing.

MSC:

12K10 Semifields
51E15 Finite affine and projective planes (geometric aspects)
51A40 Translation planes and spreads in linear incidence geometry
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[1] A. A. Albert: Finite division algebras and finite planes, Proc. Symp. Appl. Math. 10 (1960), 53–70.
[2] J. André: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Zeit. 60 (1954), 156–186. · Zbl 0056.38503 · doi:10.1007/BF01187370
[3] R. D. Baker, J. M. Dover, G. L. Ebert and K. L. Wantz: Perfect Baer subplane partitions and three-dimensional flag-transitive planes, Designs, Codes, Cryptogr. 21 (2000), 19–39. · Zbl 0970.51006 · doi:10.1023/A:1008319107762
[4] R. H. Bruck and R. C. Bose: The construction of translation planes from projective spaces, J. Algebra 1 (1964), 85–102. · Zbl 0117.37402 · doi:10.1016/0021-8693(64)90010-9
[5] R. H. Bruck and R. C. Bose: Linear representations of projective planes in projective spaces, J. Algebra 4 (1966), 117–172. · Zbl 0141.36801 · doi:10.1016/0021-8693(66)90054-8
[6] I. Cardinali, O. Polverino and R. Trombetti: Semifield planes of order q 4 with kernel \( \mathbb{F}_{q^2 } \) and center \( \mathbb{F}_q \) , European J. Combin. 27 (2006), 940–961. · Zbl 1108.51010 · doi:10.1016/j.ejc.2005.04.005
[7] P. Dembowski: Finite Geometries, Springer Verlag, Berlin, 1968. · Zbl 0159.50001
[8] G. L. Ebert, G. Marino, O. Polverino and R. Trombetti: On the multiplication of some semifields of order q 6, Finite Fields Appl. 15(2) (2009), 160–173. · Zbl 1216.12008 · doi:10.1016/j.ffa.2008.11.003
[9] G. L. Ebert, G. Marino, O. Polverino and R. Trombetti: Semifields in Class \( \mathcal{F}_4^{(a)} \) , Elect. J. Combin. 16(1) (2009), #R53 (20 pp.).
[10] H. Huang and N. L. Johnson: Semifield planes of order 82, Discrete Math. 80 (1990), 69–79. · Zbl 0699.51003 · doi:10.1016/0012-365X(90)90296-T
[11] N. L. Johnson, V. Jha and M. Biliotti: Handbook of Finite Translation Planes, Pure and Applied Mathematics, Taylor Books, 2007. · Zbl 1136.51001
[12] N. L. Johnson, G. Marino, O. Polverino and R. Trombetti: Semifields of order q 6 with left nucleus \( \mathbb{F}_{q^3 } \) and center \( \mathbb{F}_q \) , Finite Fields Appl. 14(2) (2008), 456–469. · Zbl 1137.51006 · doi:10.1016/j.ffa.2007.04.005
[13] N. L. Johnson, G. Marino, O. Polverino and R. Trombetti: On a generalization of cyclic semifields, J. Algebraic Combin. 29(1) (2009), 1–34. · Zbl 1230.51002 · doi:10.1007/s10801-007-0116-x
[14] W. M. Kantor: Commutative semifields and symplectic spreads, J. Algebra 270 (2003), 96–114. · Zbl 1041.51002 · doi:10.1016/S0021-8693(03)00411-3
[15] W. M. Kantor: Finite semifields, in: Finite Geometries, Groups, and Computation (Proc. of Conf. at Pingree Park, CO, Sept. 2005), pp. 103–114, de Gruyter, Berlin-New York, 2006. · Zbl 1102.51001
[16] D. E. Knuth: Finite semifields and projective planes, J. Algebra 2 (1965), 182–217. · Zbl 0128.25604 · doi:10.1016/0021-8693(65)90018-9
[17] D. E. Knuth: A class of projective planes, Trans. AMS 115 (1965), 541–549. · Zbl 0128.25701 · doi:10.1090/S0002-9947-1965-0202041-X
[18] G. Lunardon: Translation ovoids, J. Geom. 76 (2003), 200–215. · Zbl 1042.51008
[19] G. Lunardon: Symplectic spreads and finite semifields, Designs, Codes, Cryptogr. 44 (2007), 39–48. · Zbl 1123.51011 · doi:10.1007/s10623-007-9054-9
[20] G. Marino, O. Polverino and R. Trombetti: On \( \mathbb{F}_q \) -linear sets of PG(3,q 3) and semifields, J. Combin. Theory Ser. A 114 (2007), 769–788. · Zbl 1118.51006 · doi:10.1016/j.jcta.2006.08.012
[21] G. Marino, O. Polverino and R. Trombetti: On semifields of type (q 2n ,q n,q 2, q 2,q), n odd; Innov. Incidence Geom. 6–7 (2009), 271–290.
[22] O. Ore: On a special class of polynomials, Trans. Amer. Math. Soc. 35 (1933), 559–584. · JFM 59.0163.02 · doi:10.1090/S0002-9947-1933-1501703-0
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