Abstract
A parallelism in \(PG(n,q)\) is transitive if it has an automorphism group which is transitive on the spreads. A parallelism is regular if all its spreads are regular. In \(PG(3,4)\) no examples of transitive and no regular parallelisms are known. Transitive parallelisms in \(PG(3,4)\) must have automorphisms of order 7. That is why we construct all 482 parallelisms with automorphisms of order 7 and establish that there are neither transitive, nor regular ones among them. We conclude that there are no transitive parallelisms in \(PG(3,4)\). The investigation is computer-aided. We use GAP (Groups, Algorithms, Programming—a System for Computational Discrete Algebra) to find a subgroup of order 7 and its normalizer in the automorphism group of \(PG(3,4)\). For all the other constructions and tests we use our own software written in C++.
Similar content being viewed by others
References
Baker, R.: Partitioning the planes of \(AG_{2m}(2)\) into 2-designs. Discret. Math. 15, 205–211 (1976)
Beutelspacher, A.: On parallelisms in finite projective spaces. Geom. Dedic. 3(1), 35–45 (1974)
Bruck, R.: Construction problems of finite projective planes. In: Proceedings Conference Combinatorics, University of North Carolina Press, pp. 427–514 (1967)
Denniston, R.: Packings of PG(3, q), Finite Geometric Structures and Their Applications, Edizioni Cremonese, Rome, pp. 193–199 (1973)
Denniston, R.: Cyclic packings of the projective space of order 8. Atti Accad. Naz. Lincei Rend. 54, 373–377 (1973)
Eisfeld, J., Storme, L.: (Partial) t-spreads and minimal t-covers in finite projective spaces. Lecture notes from the Socrates Intensive Course on Finite Geometry and its Applications, Ghent, April (2000)
Fuji-Hara, R.: Mutually 2-orthogonal resolutions of finite projective space. Ars Comb. 21, 163–166 (1986)
GAP—Groups, Algorithms, Programming—a System for Computational Discrete Algebra (http://www.gap-system.org/)
Johnson, N.: Parallelisms of projective spaces. J. Geom. 76, 110–182 (2003)
Johnson, N., Montinaro, A.: The doubly transitive t-parallelisms. Results Math. 52, 75–89 (2008)
Johnson, N.: Combinatorics of Spreads and Parallelisms. CRC Press, Boca Raton (2010)
Johnson, N., Montinaro, A.: The transitive t-parallelisms of a finite projective space. Adv. Geom. 12, 401–429 (2012)
Kaski, P., Östergård, P.: Classification Algorithms for Codes and Designs. Springer, Berlin (2006)
Ledermann, W., Weir, A.: Group Theory, 2nd edn. Longman, London (1996)
Penttila, T., Williams, B.: Regular packings of PG(3, q). Eur. J. Combin. 19(6), 713–720 (1998)
Prince, A.: Parallelisms of PG(3,3) invariant under a collineation of order 5. In: Johnson, NL (ed.) Mostly Finite Geometries (Iowa City, 1996), Lect Notes Pure Appl 190, Marcel Dekker, New York, pp. 383–390 (1997)
Prince, A.: The cyclic parallelisms of PG(3,5). Eur. J. Combin. 19(5), 613–616 (1998)
Prince, A.: Covering sets of spreads in \(PG(3, q)\). Discret. Math. 238, 131–136 (2001)
Sarmiento, J.: Resolutions of PG(5,2) with point-cyclic automorphism group. J. Comb. Des. 8(1), 2–14 (2000)
Sarmiento, J.: On point-cyclic resolutions of the 2-(63,7,15) design associated with PG(5,2). Gr. Comb. 18(3), 621–632 (2002)
Silberstein, N., Etzion, T.: Codes and designs related to lifted MRD codes. In: IEEE International Symposium on Information Theory Proceedings (ISIT), pp. 2288–2292. St. Petersburg (2011)
Silberstein, N.: Coding Theory and Projective Spaces, PhD thesis, Israel Institute of Technology, Haifa (2011)
Soicher, L.: Computation of Partial Spreads, web preprint, http://www.maths.qmul.ac.uk/~leonard/partialspreads, 2000
Stinson, D., Vanstone, S.: Orthogonal packings in PG(5,2). Aequationes Math. 31(1), 159–168 (1986)
Stinson, D.: Combinatorial Designs: Constructions and Analysis. Springer, New York (2004)
Storme, L.: Finite Geometry, The CRC Handbook of Combinatorial Designs, second edition, pp. 702–729 . CRC Press, Boca Raton (2006)
Topalova, S., Zhelezova, S.: 2-spreads and transitive and orthogonal 2-parallelisms of PG(5,2). Gr. Comb. 26(5), 727–735 (2010)
Zaicev, G., Zinoviev, V., Semakov, N.: Interrelation of preparata and hamming codes and extension of hamming codes to new double-error-correcting codes, In: Proceedings of Second International Symposium on Information Theory, (Armenia, USSR, 1971), Budapest. Academiai Kiado, pp. 257–263 (1973)
Zhelezova, S.: Cyclic parallelisms of PG(5,2). Math. Balkanica 24(1–2), 141–146 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the Bulgarian National Science Fund under Contract No I01/0003.
Rights and permissions
About this article
Cite this article
Topalova, S., Zhelezova, S. On transitive parallelisms of \(PG(3,4)\) . AAECC 24, 159–164 (2013). https://doi.org/10.1007/s00200-013-0194-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-013-0194-z