Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry
S. De Winter
Ghent University Department of Pure Mathematics and Computer Algebra Krijgslaan 281-S22 B-9000 Gent Belgium Krijgslaan 281-S22 B-9000 Gent Belgium
DOI: 10.1007/s10801-006-0019-2
Abstract
Let S {\mathcal{S}} be a proper partial geometry pg( s, t,2), and let G be an abelian group of automorphisms of S {\mathcal{S}} acting regularly on the points of S {\mathcal{S}}. Then either t\equiv 2\pm od s+1 or S {\mathcal{S}} is a pg(5,5,2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63-73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.
Pages: 285–297
Keywords: keywords partial geometry; abelian singer group; geometry of Van lint-schrijver
Full Text: PDF
References
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3. R.C. Bose, “Strongly regular graphs, partial geometries and partially balanced designs,” Pacific J. Math. 13 (1963), 389-419.
4. A. Cossu, “Su alcune proprietá dei {k, n}-archidi un piano proiettivo sopra un corpo finito,” Rend. Mat. Appl. 20 (1961), 271-277.
5. F. De Clerck, M. Delanote, N. Hamilton, and R. Mathon, “Perp-systems and partial geometries,” Adv. Geom. 2 (2002), 1-12.
6. S. De Winter, “Elation and translation semipartial geometries,” J. Combin. Theory Ser. A 108(2) (2004), 313-330.
7. S. De Winter and K. Thas, “Generalized quadrangles with an abelian Singer group,” Des Codes Cryptogr. 39(1) (2006) 81-87.
8. D. Ghinelli, “Regular groups on generalized quadrangles and nonabelian difference sets with multiplier - 1.” Geom. Dedicata 41 (1992), 165-174.
9. D. Ghinelli and D. Jungnickel, “Finite projective planes with a large abelian group,” in Surveys in Combinatorics 2003, London Mathematical Society, Lecture Notes Series 307,
2003. Springer
10. W. Haemers and J.H. van Lint, “A partial geometry pg(9, 8, 4),” Ann. Discrete Math. 15 (1982), 205- 212.
11. N. Jacobson, Basic Algebra II, W. H. Freeman and Company, 1980.
12. J.H. van Lint and A. Schrijver, “Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields,” Combinatorica 1 (1981), 63-73.
13. S.L. Ma, “Regular automorphism groups on partial geometries,” Shanghai Conference Issue on Designs, Codes, and Finite Geometries, Part 1 (Shanghai, 1993). J. Statist. Plann. Inference 51(2) (1996), 215- 222.
14. S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Research Notes in Mathematics, 110, Pitman (Advanced Publishing Program), Boston MA, 1984, vi+312 pp.
15. J.A. Thas, “Some results concerning {(q + 1)(n - 1); n}-arcs and {qn - q + n; n}-arcs in finite projective planes of order q,” J. Combin. Theory Ser. A 19 (1975), 228-232.
16. J.A. Thas, “Semipartial geometries and spreads of classical polar spaces,” J. Combin. Theory Ser. A 35 (1983), 58-66.