Abstract
This paper tabulates the results of a number of computer searches in projective planes of order 16. Maximal arcs of degree 4 are found in all but two of the known planes of order 16 (and their duals). Any such arc yields a resolvable 2-(52, 4, 1) design that admits at least 52 resolutions. Pairs of disjoint degree 4 maximal arcs are also shown to exist in certain of the planes giving rise to 104-sets of type (4, 8).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.C. Bose andA. Barlotti, Linear representation of a class of projective planes in a four dimensional representation,Ann. Mat. Pura Appl.(4) 88 (1971), 9–31.
P.Bousfield, N.Hamilton, T.Penttila and G.Royle, A compendium of examples of sets of type (m, n), in preparation.
W.Bosma and J.Cannon, A language for computer algebra, The School of Mathematics, University of Sydney.
W.M. Cherowitzo, Hyperoval in translation planes of order 16,J. Combin. Math. Combin. Comput. 9 (1991), 39–55.
U. Dempwolff andA. Reifart, The translation planes of order 16 admitting a Baer 4-group,J. Comb. Theory Ser. A 32 (1982), 119–124.
U. Dempwolff andA. Reifart, The classification of translation planes of order 16.,Arch. Math. (Basel) 43 (1984), 285–288.
R.H.F. Denniston, Some maximal arcs in finite projective planes,J. Combin. Theory 6 (1969), 317–319.
M.J. De Resmini, An infinite family of type (m, n) sets in PG(2, q2),q a square,J. Geom. 20 (1983), 36–43.
N. Hamilton, Some maximal arcs in Hall planes,J. Geom. 52 (1995), 101–107.
N. Hamilton, Some inherited maximal arcs in derived dual translation planes,Geom. Dedicata 55 (1995), 165–173.
N.Hamilton,Maximal arcs in finite projective planes and associated structures in projective spaces, PhD Thesis, Univ. Western Australia, 1995.
N. Hamilton, Some maximal arcs in derived dual Hall planes,European J. Combin. 15 (1994), 525–532.
N.L. Johnson, A note on semi-translation planes of class 1-5a,Arch. Math. 21 (1970), 528–532.
N.L. Johnson, A note on the derived semifield planes of order 16,Aequationes Math. 18 (1978), 103–111.
S. Kageyama andY. Miao, Note on a paper “1-Rotational designs with block size 4”,Bull. Inst. Combin. Appl. 20 (1997), 82–84.
C.Lam, Y.Miao. and M.Mishima, Cyclically resolvable cyclic Steiner 2-systems S(2, 4, 52),J. Statist. Plann. Inference, to appear.
R.Mathon, private communication.
T. Penttila, G.F. Royle andM.K. Simpson, Hyperovals in known planes of order 16,J. Combin. Des. 4 (1996), 59–65.
S.Stoichev and V.D.Tonchev, Unital designs in planes of order 16,Appl. Discr. Math. Theoret. Comput. Sci., to appear.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hamilton, N., Stoichev, S.D. & Tonchev, V.D. Maximal arcs and disjoint maximal arcs in projective planes of order 16. J Geom 67, 117–126 (2000). https://doi.org/10.1007/BF01220304
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01220304