Cyclic 2-spreads in \(V(6, q)\) and flag-transitive linear spaces. (English) Zbl 07905710
Summary: In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open cases in the classification of flag-transitive linear spaces. We utilise the polynomial approach innovated by M. Pauley and J. Bamberg [Finite Fields Appl. 14, No. 2, 537–548 (2008; Zbl 1137.51007)] to obtain our results.
MSC:
| 51E20 | Combinatorial structures in finite projective spaces |
| 51E23 | Spreads and packing problems in finite geometry |
| 51A40 | Translation planes and spreads in linear incidence geometry |
| 05B25 | Combinatorial aspects of finite geometries |
| 11T06 | Polynomials over finite fields |
Citations:
Zbl 1137.51007References:
| [1] | Aubry, Y.; Perret, M., A Weil theorem for singular curves, (Pellikaan, R.; Perret, M.; Vlăduţ, S. G., Arithmetic, Geometry, and Coding Theory, 1996, De Gruyter: De Gruyter Berlin, New York), 1-8 · Zbl 0873.11037 |
| [2] | Barlotti, A.; Cofman, J., Finite Sperner spaces constructed from projective and affine spaces, Abh. Math. Semin. Univ. Hamb., 40, 231-241, 1974 · Zbl 0271.50007 |
| [3] | Bartoli, D.; Timpanella, M., A family of permutation trinomials over \(\mathbb{F}_{q^2} \), Finite Fields Appl., 70, Article 101781 pp., 2021 · Zbl 1472.11295 |
| [4] | Blake, I. F.; Gao, S.; Mullin, R. C., Normal and self-dual normal bases from factorization of \(c x^{q + 1} + d x^q - a x - b\), SIAM J. Discrete Math., 7, 3, 499-512, 1994 · Zbl 0806.11061 |
| [5] | Buekenhout, F.; Delandtsheer, A.; Doyen, J., Finite linear spaces with flag-transitive groups, J. Comb. Theory, Ser. A, 49, 2, 268-293, 1988 · Zbl 0658.20001 |
| [6] | Buekenhout, F.; Delandtsheer, A.; Doyen, J.; Kleidman, P. B.; Liebeck, M. W.; Saxl, J., Linear spaces with flag-transitive automorphism groups, Geom. Dedic., 36, 1, 89-94, Oct 1990 · Zbl 0707.51017 |
| [7] | Dickson, L. E., Criteria for the irreducibility of functions in a finite field, Bull. Am. Math. Soc., 13, 1, 1-8, 1906 · JFM 37.0094.01 |
| [8] | Feng, T.; Lu, J., New families of flag-transitive linear spaces, Finite Fields Appl., 87, Article 102156 pp., 2023 · Zbl 1511.05027 |
| [9] | Gow, R.; McGuire, G., Invariant rational functions, linear fractional transformations and irreducible polynomials over finite fields, Finite Fields Appl., 79, Article 101991 pp., 2022 · Zbl 1484.12004 |
| [10] | Higman, D. G.; McLaughlin, J. E., Geometric ABA-groups, Ill. J. Math., 5, 3, 382-397, 1961 · Zbl 0104.14702 |
| [11] | Kantor, W. M., 2-transitive and flag-transitive designs, (Coding Theory, Design Theory, Group Theory. Coding Theory, Design Theory, Group Theory, Burlington, VT, 1990. Coding Theory, Design Theory, Group Theory. Coding Theory, Design Theory, Group Theory, Burlington, VT, 1990, Wiley-Intersci. Publ., 1993, Wiley: Wiley New York), 13-30 |
| [12] | Lidl, R.; Niederreiter, H., Finite Fields, Encyclopedia of Mathematics and Its Applications, 1996, Cambridge University Press · Zbl 0866.11069 |
| [13] | Liebeck, M. W., The classification of finite linear spaces with flag-transitive automorphism groups of affine type, J. Comb. Theory, Ser. A, 84, 2, 196-235, 1998 · Zbl 0918.51009 |
| [14] | Ore, O., Contributions to the theory of finite fields, Trans. Am. Math. Soc., 36, 2, 243-274, 1934 · JFM 60.0111.04 |
| [15] | Pauley, M.; Bamberg, J., A construction of one-dimensional affine flag-transitive linear spaces, Finite Fields Appl., 14, 2, 537-548, 2008 · Zbl 1137.51007 |
| [16] | Saxl, J., On finite linear spaces with almost simple flag-transitive automorphism groups, J. Comb. Theory, Ser. A, 100, 2, 322-348, 2002 · Zbl 1031.51009 |
| [17] | Segre, B., Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64, 1-76, 1964 · Zbl 0128.15002 |
| [18] | Stichtenoth, H.; Topuzoğlu, A., Factorization of a class of polynomials over finite fields, Finite Fields Appl., 18, 1, 108-122, 2012 · Zbl 1263.12002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
