Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T08:13:50.718Z Has data issue: false hasContentIssue false
  • English
  • Français

COORDINATISING PLANES OF PRIME POWER ORDER USING FINITE FIELDS

Part of: Designs and configurations Finite geometry and special incidence structures General field theory

Published online by Cambridge University Press:  22 August 2018

ROBERT S. COULTER Open the ORCID record for ROBERT S. COULTER [Opens in a new window]
ROBERT S. COULTER*
Affiliation:
520 Ewing Hall, Department of Mathematical Sciences, University of Delaware, Newark, DE19716, USA email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We revisit the coordinatisation method for projective planes by considering the consequences of using finite fields to coordinatise projective planes of prime power order. This leads to some general restrictions on the form of the resulting planar ternary ring (PTR) when viewed as a trivariate polynomial over the field. We also consider how the Lenz–Barlotti type of the plane being coordinatised impacts the form of the PTR polynomial, thereby deriving further restrictions.

Keywords

projective planespolynomials over finite fieldsplanar ternary ringscoordinatisation method

MSC classification

Primary: 51E15: Affine and projective planes
Secondary: 12E05: Polynomials (irreducibility, etc.) 05B25: Finite geometries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 106 , Issue 2 , April 2019 , pp. 184 - 199
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The results of this article were presented as part of a plenary talk given at the National Conference on Coding Theory and Cryptography 2017 (1–4 September), in Hangzhou, China.

References

Barlotti, A., ‘Le possibili configurazioni del sistema delle coppie punto-retta (A, a) per cui un piano grafico risulta (A, a)-transitivo’, Boll. Unione Mat. Ital. (9) 12 (1957), 212–226.Google Scholar
Castillo, C. and Coulter, R. S., ‘A general representation theory for constructing groups of permutation polynomials’, Finite Fields Appl. 35 (2015), 172–203.Google Scholar
Dembowski, P., Finite Geometries (Springer, Berlin, 1968), reprinted 1997.Google Scholar
Dickson, L. E., ‘On commutative linear algebras in which division is always uniquely possible’, Trans. Amer. Math. Soc. 7 (1906), 514–522.Google Scholar
Ghinelli, D. and Jungnickel, D., ‘On finite projective planes in Lenz–Barlotti class at least I.3’, Adv. Geom. (Suppl) (2003), S28–S48.Google Scholar
Hall, M., ‘Projective planes’, Trans. Amer. Math. Soc. 54 (1943), 229–277.Google Scholar
Hering, C. H. and Kantor, W. M., ‘On the Lenz–Barlotti classification of projective planes’, Arch. Math. 22 (1971), 221–224.Google Scholar
Hughes, D. R. and Piper, F. C., Projective Planes, Graduate Texts in Mathematics, 6 (Springer, New York, 1973).Google Scholar
Lam, C. W. H., Thiel, L. and Swiercz, S., ‘The non-existence of finite projective planes of order 10’, Canad. J. Math. 41 (1989), 1117–1123.Google Scholar
Lenz, H., ‘Zur Begründung der analytischen Geometrie’, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. (1954), 17–72.Google Scholar
Lidl, R. and Mullen, G. L., ‘When does a polynomial over a finite field permute the elements of the field?’, Amer. Math. Monthly 95 (1988), 243–246.Google Scholar
Lüneberg, H., ‘Zur Frage der Existenz von endlichen projektiven Ebenen vom Lenz–Barlotti-Typ III.2’, J. reine angew. Math. 220 (1965), 63–67.Google Scholar
Matthews, R. W., ‘Permutation polynomials in one and several variables’, PhD Thesis, University of Tasmania, Hobart, 1990.Google Scholar
Mullen, G. L. and Panario, D., Handbook of Finite Fields, Discrete Mathematics and Its Applications, 78 (CRC Press, Boca Raton, FL, 2013).Google Scholar
Yaqub, J. C. D. S., ‘The non-existence of finite projective planes of Lenz–Barlotti class III.2’, Arch. Math. 18 (1967), 308–312.Google Scholar