Towards the determination of the regular $n$-covers of $PG(3,q)$

Oxenham, Martin; Casse, Rey

Towards the determination of the regular

n

-covers of

P G (3, q)

Oxenham, Martin ; Casse, Rey

Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003), p. 57-87 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Abstract
Résumé

A set of lines $S$ of $P G (3, q)$ is said to cover a point $P$ of $P G (3, q)$ $n$ times if there are exactly $n$ lines of $S$ incident with $P$ . An $n$ -cover of $P G (3, q)$ is a set of lines of $P G (3, q)$ which covers each point of $P G (3, q)$ $n$ times. In this paper, the properties and known examples of $n$ -covers are reviewed and it is demonstrated how $n$ -covers of $P G (3, q)$ can be used to construct classes of quasi- $n$ -multiple Sperner designs. Finally, motivated by the problem of deriving these designs to arrive at new examples, the notion of regular $n$ -covers of $P G (3, q)$ is introduced. The main results of the paper are that no regular $2$ -covers of $P G (3, q)$ exist for $q > 2$ and that no regular $n$ -covers $(n \geq 3)$ exist whenever $q \geq n + 2$ .

Si dice che un insieme $S$ di rette di $P G (3, q)$ copre $n$ volte un punto $P$ di $P G (3, q)$ , se esistono esattamente $n$ rette di $S$ incidenti $P$ . Un insieme di rette di $P G (3, q)$ che copre $n$ volte ogni punto di $P G (3, q)$ si dice $n$ -cover. In questa nota, dopo una descrizione degli esempi noti di $n$ -cover e delle rispettive proprietà, viene mostrato come gli $n$ -cover di $P G (3, q)$ possono essere utilizzati per la costruzione di classi di disegni di Sperner quasi- $n$ -multipli. Infine, allo scopo di ottenere nuovi esempi di tali disegni mediante la derivazione di quelli esistenti, si introduce la nozione di n-cover regolare. I risultati principali sono: la dimostrazione della non esistenza di un $2$ -cover regolare di $P G (3, q)$ per $q > 2$ e quella della non esistenza di un $n$ -cover regolare $(n \geq 3)$ per $q \geq n + 2$ .

Publié le : 2003-02-01

MR 1955697 | Zbl 1177.51007

@article{BUMI_2003_8_6B_1_57_0,
     author = {Martin Oxenham and Rey Casse},
     title = {Towards the determination of the regular $n$-covers of $PG(3,q)$},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {6-A},
     year = {2003},
     pages = {57-87},
     zbl = {1177.51007},
     mrnumber = {1955697},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2003_8_6B_1_57_0}
}

Oxenham, Martin; Casse, Rey. Towards the determination of the regular $n$-covers of $PG(3,q)$. Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003) pp. 57-87. http://gdmltest.u-ga.fr/item/BUMI_2003_8_6B_1_57_0/

Bibliographie

[1] Bagchi, B.-Sastry, N. S. M., Even order inversive planes, generalized quadrangles and codes, Geom. Dedicata, 22 (1987), 137-147. | MR 877206 | Zbl 0609.51011

[2] Bagchi, B.-Sastry, N. S. N., Intersection patterns of the classical ovoids in symplectic 3-space of even order, J. Algebra, 126 (1989), 147-160. | MR 1023290 | Zbl 0685.51006

[3] Barlotti, A., SOME TOPICS IN FINITE GEOMETRICAL STRUCTURESInstitute of Statistics Mimeo Series, 439, University of North Carolina, 1965.

[4] Beutelspacher, A., On parallelisms in finite projective spaces, Geom. Dedicata., 3 (1974), 35-40. | MR 341270 | Zbl 0282.50019

[5] Beutelspacher, A., On $t$ -covers in finite projective spaces, J. Geom., 12, no. 1 (1979), 10-16. | MR 521135 | Zbl 0369.05020

[6] Billington, E. J., CONSTRUCTION OF SOME IRREDUCIBLE DESIGNS, Lecture Notes in Mathematics, Springer-Verlag, 1981. | MR 674137 | Zbl 0488.05010

[7] Billington, E. J., Further constructions of irreducible designs, Congr. Numer., 35 (1982), 77-79. | MR 725870 | Zbl 0513.05010

[8] Bose, R. C.-Bruck, R. H., The construction of translation planes from projective spaces, J. Algebra, 1 (1964), 85-102. | MR 161206 | Zbl 0117.37402

[9] Bruck, R. H., Construction problems of finite projective planes, in: Combinatorial Mathematics and its Applications, University of North Carolina Press, Chapel Hill, 1969. | MR 250182 | Zbl 0206.23402

[10] Bruen, A. A.-Fisher, J. C., Spreads which are not dual spreads, Canad. Math. Bull., 12, no. 6 (1969). | MR 256257 | Zbl 0186.54303

[11] Cameron, P. J.-Van Lint, J. H., Graphs, codes and designs, London Mathematical Society Lecture Note Series, 43, Cambridge University Press. | MR 579788 | Zbl 0427.05001

[12] Denniston, R. H. F., Some packings of projective spaces, Atti Accad. Naz. Lincei Rend., 52 (1972), 36-40. | MR 331207 | Zbl 0239.50013

[13] Denniston, R. H. F., Cyclic packings of the projective space of order 8, Atti Accad. Naz. Lincei Rend., 54 (1973), 373-377. | MR 362028 | Zbl 0307.50017

[14] Ebert, G. L., Partitioning projective geometries into caps, Canad. J. Math., 37, no. 6 (1985), 1163-1175. | MR 828840 | Zbl 0571.51002

[15] Ebert, G. L., The completion problem for partial packings, Geom. Dedicata, 18 (1985), 261-267. | MR 797145 | Zbl 0566.51013

[16] Ebert, G. L., Spreads obtained from ovoidal fibrations, in: Finite Geometries, Lecture Notes In Pure And Applied Mathematics, 103, Marcel Dekker, 1985. | MR 826801 | Zbl 0577.51005

[17] Foulser, D. A., Replaceable translation nets, Proc. London Math. Soc., 22 (3) (1971), 235-264. | MR 291935 | Zbl 0212.52303

[18] Glynn, D., On a set of lines of $P G (3, q)$ corresponding to a maximal cap contained in the Klein quadric of $P G (5, q)$ , Geom. Dedicata, 26, no. 3 (1988). | MR 950065 | Zbl 0645.51012

[19] Hall, M. Jr., Combinatorial theory, Blaisdell Publishing Company, 1967. | MR 224481 | Zbl 0196.02401

[20] Hall, M. Jr., Incidence axioms for affine geometries, J. Algebra, 21 (1972), 535-547. | MR 317160 | Zbl 0252.50010

[21] Hill, R., On Pellegrino's $20$ -Caps In $S_{4, 3}$ , Annals of Discrete Mathematics, 18 (1983), 433-448. | MR 695829 | Zbl 0505.51013

[22] Hilton, A. J. W.-Tierlinck, L., Dimension in Steiner triple systems, Ann. Discrete Math., 7 (1980), 73-87. | MR 584405 | Zbl 0441.05011

[23] Hirschfeld, J. W. P., Projective geometries over finite fields, Clarendon Press, Oxford. | MR 1612570 | Zbl 0899.51002

[24] Hirschfeld, J. W. P., Finite projective spaces of three dimensions, Clarendon Press, Oxford, 1985. | MR 840877 | Zbl 0574.51001

[25] Jessop, C. M., The line complex, Chelsea, 1969.

[26] Jungnickel, D., Quasimultiples of biplanes and residual biplanes, Ars Combin., 19 (1984), 179-186. | MR 810274 | Zbl 0572.05011

[27] Jungnickel, D., Quasimultiples of projective and affine planes, J. Geom., 26 (1986). | MR 850162 | Zbl 0586.51006

[28] Kramer, E. S., Indecomposable triple systems, Discrete Math., 8 (1974), 173-180. | MR 332536 | Zbl 0276.05020

[29] Mathon, R.-Rosa, A., A census of Mendelsohn triple systems, Ars Combin., 4 (1977), 309-315. | MR 462968 | Zbl 0442.05006

[30] Mathon, R.-Rosa, A., Tables of parameters of BIBDS with $r \leq 41$ including existence, enumeration and resolvability results, Ann. Discrete Math., 26 (1985), 275-308. | MR 833795 | Zbl 0579.05016

[31] Morgan, E. J., Some small quasi-multiple designs, Ars Combin., 3 (1977), 233-250. | MR 457248 | Zbl 0394.05005

[32] Morgan, E. J., Balanced ternary designs with block size three, in: Combinatorial Mathematics VII, Lecture Notes in Mathematics, 829, Springer-Verlag, 1980. | MR 611194 | Zbl 0454.05017

[33] Nicoletti, G., Su una nuova classe di spazi affini generalizzati di Sperner, Atti Accad. Naz. Lincei Rend., 59 (1975). | MR 487751 | Zbl 0342.50013

[34] Orr, W. F., A characterization of subregular spreads in finite $3$ -space, Geom. Dedicata, 5 (1976), 43-50. | MR 470834 | Zbl 0335.50012

[35] Oxenham, M. G.-Casse, L. R. A., On a Geometric Representation of the Subgroups of Index $6$ in $S_{6}$ , Discrete Mathematics, 92 (1991), 251-259. | MR 1140591 | Zbl 0752.20002

[36] Oxenham, M. G., On $n$ -Covers of $P G (3, q)$ and Related Structures, Doctoral Thesis, University of Adelaide, 1991.

[37] Oxenham, M. G.-Casse, L. R. A., On the Resolvability of Hall Triple Systems, Bolletino U.M.I., 8, 1-B (1998), 639-649. | MR 1662345 | Zbl 0918.05019

[38] Payne, S. E.-Thas, J. A., Finite Generalized Quadrangles, Research Notes in Mathematics, 110, Pitman Advanced Publishing Program, 1984. | MR 767454 | Zbl 0551.05027

[39] Pellegrino, G., Sulle Calotte Massime Dello Spazio $S_{4, 3}$ , Atti dell'Accad. di Scienze lettere e Arti di Palermo, Serie IV, Vol. XXXIV (1974-75), 297-328. | MR 465903 | Zbl 0443.51008

[40] Penttila, T., PRIVATE COMMUNICATION, 1990.

[41] Penttila, T.-Williams, B., Regular Packings of $P G (3, q)$ , Europ. J. Combinatorics, 19 (1998), 713-720. | MR 1642722 | Zbl 0920.51006

[42] Prince, A. R., The Cyclic Parallelisms of $P G (3, q)$ , Europ. J. Combinatorics, 19 (1998), 613-616. | MR 1637760 | Zbl 0907.51004

[43] Rao, C. R., Cyclical generation of linear subspaces in finite geometries, Conference on Combinatorial Mathematics and its Applications, (University of North Carolina, 1967), University of North Carolina Press, 1969, 515-535. | MR 249317 | Zbl 0211.53203

[44] Samărdziski, A., A class of finite Sperner spaces, Abh. Math. Sem. Univ. Hamburg, 42 (1974). | MR 358538 | Zbl 0295.50031

[45] Segre, B., Teoria di Galois, fibrazioni proiettive e geometrie non Desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76. | MR 169117 | Zbl 0128.15002

[46] Singer, J., A theorem in finite geometry and some applications to number theory, Trans. Amer. Math. Soc., 43 (1938), 377-385. | JFM 64.0972.04 | MR 1501951

[47] Sylvester, J. J., Collected mathematical papers, I. Cambridge University Press, (1904). | JFM 35.0020.01

[48] Teirlinck, L., On linear spaces in which every plane is either projective or affine, Geom. Dedicata, 4 (1975), 39-44. | MR 384567 | Zbl 0309.50014

[49] Teirlinck, L., Combinatorial properties of planar spaces and embeddability, J. Combin. Theory A., 43 (1986), 291-302. | MR 867653 | Zbl 0605.51009

[50] Van Maldeghem, H., Generalized Polygons, Monographs in Mathematics, Birkhäuser, 1998. | MR 1725957 | Zbl 0914.51005

[51] Wertheimer, M. A., A double affine plane of order $6$ , J. Combin. Theory A., 56 (1991), 166-171. | MR 1082850 | Zbl 0748.05023