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COORDINATISING PLANES OF PRIME POWER ORDER USING FINITE FIELDS

  • ROBERT S. COULTER (a1)

Abstract

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.

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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.

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[1] 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), 212226.
[2] Castillo, C. and Coulter, R. S., ‘A general representation theory for constructing groups of permutation polynomials’, Finite Fields Appl. 35 (2015), 172203.
[3] Dembowski, P., Finite Geometries (Springer, Berlin, 1968), reprinted 1997.
[4] Dickson, L. E., ‘On commutative linear algebras in which division is always uniquely possible’, Trans. Amer. Math. Soc. 7 (1906), 514522.
[5] Ghinelli, D. and Jungnickel, D., ‘On finite projective planes in Lenz–Barlotti class at least I.3’, Adv. Geom. (Suppl) (2003), S28S48.
[6] Hall, M., ‘Projective planes’, Trans. Amer. Math. Soc. 54 (1943), 229277.
[7] Hering, C. H. and Kantor, W. M., ‘On the Lenz–Barlotti classification of projective planes’, Arch. Math. 22 (1971), 221224.
[8] Hughes, D. R. and Piper, F. C., Projective Planes, Graduate Texts in Mathematics, 6 (Springer, New York, 1973).
[9] Lam, C. W. H., Thiel, L. and Swiercz, S., ‘The non-existence of finite projective planes of order 10’, Canad. J. Math. 41 (1989), 11171123.
[10] Lenz, H., ‘Zur Begründung der analytischen Geometrie’, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. (1954), 1772.
[11] 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), 243246.
[12] Lüneberg, H., ‘Zur Frage der Existenz von endlichen projektiven Ebenen vom Lenz–Barlotti-Typ III.2’, J. reine angew. Math. 220 (1965), 6367.
[13] Matthews, R. W., ‘Permutation polynomials in one and several variables’, PhD Thesis, University of Tasmania, Hobart, 1990.
[14] Mullen, G. L. and Panario, D., Handbook of Finite Fields, Discrete Mathematics and Its Applications, 78 (CRC Press, Boca Raton, FL, 2013).
[15] Yaqub, J. C. D. S., ‘The non-existence of finite projective planes of Lenz–Barlotti class III.2’, Arch. Math. 18 (1967), 308312.
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COORDINATISING PLANES OF PRIME POWER ORDER USING FINITE FIELDS

  • ROBERT S. COULTER (a1)

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