×

zbMATH — the first resource for mathematics

Commutative semifields and symplectic spreads. (English) Zbl 1041.51002
Recently, much attention is again being paid to semi-fields; that is, algebras satisfying all of the axions for a skew field except (possibly) associativity. One of the earliest papers on semi-fields was [D.E. Knuth, J. Algebra 2, 182-217 (1965; Zbl 0128.25604)], in which Donald Knuth defined objects called cubical arrays.
In the paper under review the author uses these cubical arrays to provide a bijection between commutative semi-field planes and symplectic semi-field planes. While the number of previously known commutative semi-field planes of any given order \(N\) was less than log \(N\), the number constructed in this remarkable paper is not bounded above by any polynomial in \(N\), where here the order \(N\) of the planes constructed is even. Thus, the gauntlet has been thrown down, and there are now many different (mutually non-isomorphic) commutative semi-field planes known in characteristic 2, but not nearly so many in odd characteristic. However, as the author points out, there are more types of constructions for commutative semi-fields in odd characteristic, and one still would like more examples in characteristic 2 having odd dimension over GF(2) because of the connection with coding theory.

MSC:
51A40 Translation planes and spreads in linear incidence geometry
51A35 Non-Desarguesian affine and projective planes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albert, A.A., On nonassociative division algebras, Trans. amer. math. soc., 72, 296-309, (1952) · Zbl 0046.03601
[2] Albert, A.A., Finite division algebras and finite planes, (), 53-70
[3] Albert, A.A., On the collineation groups associated with twisted fields, (), 485-497
[4] Albert, A.A., Generalized twisted fields, Pacific J. math., 11, 1-8, (1961) · Zbl 0154.27203
[5] Albert, A.A., Isotopy for generalized twisted fields, An. acad. brasil. ciênc., 33, 265-275, (1961) · Zbl 0154.27301
[6] S. Ball, M.R. Brown, The six semifield planes associated with a semifield flock, Preprint · Zbl 1142.12305
[7] Biliotti, M.; Jha, V.; Johnson, N., The collineation groups of generalized twisted field planes, Geom. dedicata, 76, 97-126, (1999) · Zbl 0936.51003
[8] Bader, L.; Kantor, W.M.; Lunardon, G., Symplectic spreads from twisted fields, Boll. un. mat. ital., 8-A, 383-389, (1994) · Zbl 0827.51005
[9] Ball, S.; Lavrauw, M., Commutative semifields of rank 2 over their middle nucleus, (), 1-21 · Zbl 1102.12300
[10] Bader, L.; Lunardon, G.; Pinneri, J., A new semifield flock, J. combin. theory ser. (A), 86, 49-62, (1999) · Zbl 0937.51005
[11] Burmester, M.V.D., On the commutative non-associative division algebras of even order of L.E. dickson, Rend. mat. appl., 21, 143-166, (1962) · Zbl 0146.26101
[12] Calderbank, A.R.; Cameron, P.J.; Kantor, W.M.; Seidel, J.J., \(Z4\)-kerdock codes, orthogonal spreads, and extremal Euclidean line-sets, Proc. London math. soc., 75, 436-480, (1997) · Zbl 0916.94014
[13] Cohen, S.D.; Ganley, M.J., Commutative semifields, two-dimensional over their middle nuclei, J. algebra, 75, 373-385, (1982) · Zbl 0499.12021
[14] Cordero, M.; Wene, G.P., A survey of finite semifields, Discrete math., 208/209, 125-137, (1999) · Zbl 1031.12009
[15] Coulter, R.S.; Matthews, R.W., Planar functions and planes of lenz – barlotti class II, Des. codes cryptogr., 10, 167-184, (1997) · Zbl 0872.51007
[16] Dembowski, P., Finite geometries, (1968), Springer Berlin · Zbl 0159.50001
[17] Dembowski, P.; Ostrom, T.G., Planes of order n with a collineation group of order n2, Math. Z., 103, 239-258, (1968) · Zbl 0163.42402
[18] Dickson, L.E., On finite algebras, Göttingen nachrichtung, 358-393, (1905) · JFM 36.0138.03
[19] Dickson, L.E., Linear algebras in which division is always uniquely possible, Trans. amer. math. soc., 7, 370-390, (1906) · JFM 37.0111.06
[20] Dickson, L.E., On commutative linear algebras in which division is always uniquely possible, Trans. amer. math. soc., 7, 514-522, (1906) · JFM 37.0112.01
[21] Ganley, M.J., Central weak nucleus semifields, European J. combin., 2, 339-347, (1981) · Zbl 0469.51005
[22] Hughes, D.R.; Kleinfeld, E., Seminuclear extensions of Galois fields, Amer. J. math., 82, 389-392, (1960) · Zbl 0097.02201
[23] Kantor, W.M., Spreads, translation planes and kerdock sets I, II, SIAM J. alg. discr. meth., SIAM J. alg. discr. meth., 3, 308-318, (1982) · Zbl 0535.51003
[24] Kantor, W.M., Ovoids and translation planes, Canad. J. math., 34, 1195-1207, (1982) · Zbl 0467.51004
[25] Kantor, W.M., Expanded, sliced and spread spreads, (), 251-261 · Zbl 0548.51011
[26] Kantor, W.M., Codes, quadratic forms and finite geometries, (), 153-177 · Zbl 0867.94036
[27] Knuth, D.E., Finite semifields and projective planes, J. algebra, 2, 182-217, (1965) · Zbl 0128.25604
[28] Knuth, D.E., A class of projective planes, Trans. amer. math. soc., 115, 541-549, (1965) · Zbl 0128.25701
[29] W.M. Kantor, M.E. Williams, Symplectic semifield planes and \(Z4\)-linear codes, Trans. Amer. Math. Soc., in press. http://darkwing.uoregon.edu/ kantor/PAPERS/semifieldZ4Codes2.pdf · Zbl 1038.51003
[30] A. Maschietti, Symplectic translation planes and line ovals, Adv. Geom., in press · Zbl 1030.51002
[31] Menichetti, G., n-dimensional algebras over a field with a cyclic extension of degree n, Geom. dedicata, 63, 69-94, (1996) · Zbl 0869.17002
[32] Prince, A.R., Two new families of commutative semifields, Bull. London math. soc., 32, 547-550, (2000) · Zbl 1025.12004
[33] Penttila, T.; Williams, B., Ovoids of parabolic spaces, Geom. dedicata, 82, 1-19, (2000) · Zbl 0969.51008
[34] Sandler, R., The collineation groups of some finite projective planes, Portugal. math., 21, 189-199, (1962) · Zbl 0136.30304
[35] Taylor, D.E., The geometry of the classical groups, (1992), Heldermann Berlin · Zbl 0760.05056
[36] Thas, J.A.; Payne, S.E., Spreads and ovoids in finite generalized quadrangles, Geom. dedicata, 52, 227-253, (1994) · Zbl 0804.51007
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.