Abstract
Let π be ak-net of ordern with line-point incidence matrixN and letA be the adjacency matrix of its collinearity graph. In this paper we study thep-ranks (that is, the rank over\(\mathbb{F}_p \)) of the matrixA+kl withp a prime dividingn. SinceA+kI=N T N thesep-ranks are closely related to thep-ranks ofN. Using results of Moorhouse on thep-ranks ofN, we can determiner p (A+kI) if π is a 3-net (latin square) or a desarguesian net of prime order. On the other hand we show how results for thep-ranks ofA+kI can be used to get results for thep-ranks ofN, especially in connection with the Moorhouse conjecture. Finally we generalize the result of Moorhouse on thep-rank ofN for desarguesian nets of orderp a bit to special subnets of the desarguesian affine plane of orderp e.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. E. Brouwer and C. A. van Eijl, On thep-rank of the adjacency matrices of strongly regular graphs,Journal of Alg. Comb. Vol. 1 (1992) pp. 329–346.
R. H. Bruck,A Survey of Binary Systems, Springer-Verlag, Berlin (1958).
R. H. Bruck, Finite nets II, Uniqueness and embedding,Pacific J. Math. Vol. 13 (1963) pp. 421–457.
P. J. Cameron, Strongly Regular Graphs, in:Selected topics in graph theory, L. W. Beineke and R. J. Wilson (eds.), Academic Press, London, (1978) pp. 337–360.
J. Dénes and A. D. Keedwell,Latin Squares, New Developments in the Theory and Applications, Annals of Discrete Mathematics 46, North-Holland, Amsterdam (1991).
S. Dougherty, Nets and their codes,Designs, Codes and Cryptography, Vol. 3 (1993) pp. 315–331.
K. W. Johnson and J. D. H. Smith, Characters of finite quasigroups,European J. Combin. Vol. 5 (1984) pp. 43–50.
K. W. Johnson and J. D. H. Smith, Characters of finite quasigroups II: induced characters,European J. Combin. Vol. 7 (1986) pp. 131–137.
K. W. Johnson and J. D. H. Smith, Characters of finite quasigroups III: quotients and fusion,European J. Combin. Vol. 10 (1989) pp. 47–56.
K. W. Johnson and J. D. H. Smith, Characters of finite quasigroups IV: products and superschemes,European J. Combin. Vol. 10 (1989) pp. 257–263.
K. W. Johnson and J. D. H. Smith, Characters of finite quasigroups V: linear characters,European J. Combin. Vol. 10 (1989) pp. 449–456.
K. W. Johnson and J. D. H. Smith, Characters of finite quasigroups VI: critical examples and doubletons,European J. Combin. Vol. 11 (1990) pp. 267–275.
E. S. Lander,Symmetric Designs: An Algebraic Approach, Lond. Math. Soc. Lecture Notes, 74, Cambridge Univ. Press (1983).
G. E. Moorhouse, Bruck nets, codes, and characters of loops,Designs, Codes and Cryptography, Vol. 1 (1991) pp. 7–29.
Author information
Authors and Affiliations
Additional information
Communicated by: D. Jungnickel
The author is financially supported by the Cooperation Centre Tilburg and Eindhoven Universities.
Rights and permissions
About this article
Cite this article
Peeters, R. On thep-ranks of net graphs. Des Codes Crypt 5, 139–153 (1995). https://doi.org/10.1007/BF01397667
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01397667