Skip to main content



Let $\mathbb{F}_{q}^{n}$ be a vector space of dimension $n$ over the finite field $\mathbb{F}_{q}$ . A  $q$ -analog of a Steiner system (also known as a $q$ -Steiner system), denoted ${\mathcal{S}}_{q}(t,\!k,\!n)$ , is a set ${\mathcal{S}}$ of $k$ -dimensional subspaces of $\mathbb{F}_{q}^{n}$ such that each $t$ -dimensional subspace of $\mathbb{F}_{q}^{n}$ is contained in exactly one element of ${\mathcal{S}}$ . Presently, $q$ -Steiner systems are known only for $t\,=\,1\!$ , and in the trivial cases $t\,=\,k$ and $k\,=\,n$ . In this paper, the first nontrivial $q$ -Steiner systems with $t\,\geqslant \,2$ are constructed. Specifically, several nonisomorphic $q$ -Steiner systems ${\mathcal{S}}_{2}(2,3,13)$ are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of $\text{GL}(13,2)$ . This approach leads to an instance of the exact cover problem, which turns out to have many solutions.

    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Available formats
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Available formats
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Available formats
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hide All
[1] Abel, R. J. R. and Buratti, M., ‘Difference families’, inHandbook of Combinatorial Designs, 2nd edn, (eds. Colbourn, C. J. and Dinitz, J. H.) (Chapman & Hall/CRC, Boca Raton, 2007), 392410.
[2] Abel, R. J. R. and Greig, M., ‘BIBDs with small block size’, inHandbook of Combinatorial Designs, 2nd edn, (eds. Colbourn, C. J. and Dinitz, J. H.) (Chapman & Hall/CRC, Boca Raton, 2007), 7279.
[3] Ahlswede, R., Aydinian, H. K. and Khachatrian, L. H., ‘On perfect codes and related concepts’, Des. Codes Cryptogr. 22 (2001), 221237.
[4] Baer, R., Linear Algebra and Projective Geometry (Academic Press, New York, 1952).
[5] Beth, T., Jungnickel, D. and Lenz, H., Design Theory, 2nd edn, Vol. I. (Cambridge University Press, Cambridge, 1999).
[6] Beutelspacher, A., ‘Parallelismen in unendlichen projektiven Raumen endlicher Dimension’, Geom. Dedicata 7 (1978), 499506.
[7] Braun, M., Kerber, A. and Laue, R., ‘Systematic construction of q-analogs of designs’, Des. Codes Cryptogr. 34 (2005), 5570.
[8] Cameron, P., ‘Generalisation of Fisher’s inequality to fields with more than one element’, inCombinatorics, (eds. McDonough, T. P. and Mavron, V. C.) London Math. Soc. Lecture Note, 13 (Cambridge University Press, Cambridge, 1974), 913.
[9] Cameron, P., ‘Locally symmetric designs’, Geom. Dedicata 3 (1974), 6576.
[10] Cayley, A., ‘On the triadic arrangements of seven and fifteen things’, Philos. Mag. 37 (1850), 5053.
[11] Cohn, H., ‘Projective geometry over F1 and the Gaussian binomial coefficients’, Amer. Math. Monthly 111 (2004), 487495.
[12] Colbourn, C. J. and Dinitz, J. H. (Eds.), Handbook of Combinatorial Designs, 2nd edn, (Chapman & Hall/CRC, Boca Raton, 2007).
[13] Cossidente, A. and de Resmini, M. J., ‘Remarks on Singer cyclic groups and their normalizers’, Des. Codes Cryptogr. 32 (2004), 97102.
[14] Delsarte, P., ‘Association schemes and t-designs in regular semilattices’, J. Combin. Theory Ser. A 20 (1976), 230243.
[15] Dye, R. H., ‘Maximal subgroups of symplectic groups stabilizing spreads II’, J. Lond. Math. Soc. 40(2) (1989), 215226.
[16] Etzion, T., ‘Covering of subspaces by subspaces’, Des. Codes Cryptogr. 72 (2014), 405421.
[17] Etzion, T. and Vardy, A., ‘Error-correcting codes in projective space’, IEEE Trans. Inform. Theory 57 (2011), 11651173.
[18] Etzion, T. and Vardy, A., ‘On q-analogs for Steiner systems and covering designs’, Adv. Math. Commun. 5 (2011), 161176.
[19] Euler, L., ‘Consideratio quarumdam serierum quae singularibus proprietatibus sunt praeditae’, Novi Comment. Acad. Sci. Petropolitanae 3(1750–1751) 1012. 86–108; Opera Omnia, Ser. I, vol. 14, B. G. Teubner, Leipzig, 1925, pp. 516–541.
[20] Fazeli, A., Lovett, S. and Vardy, A., ‘Nontrivial t-designs over finite fields exist for all t ’, J. Combin. Theory Ser. A 127 (2014), 149160.
[21] Goldman, J. R. and Rota, G.-C., ‘On the foundations of combinatorial theory IV: finite vector spaces and Eulerian generating functions’, Stud. Appl. Math. 49 (1970), 239258.
[22] Greig, M., ‘Some balanced incomplete block design constructions’, Congr. Numer. 77 (1990), 121134.
[23] Hanani, H., ‘A class of three-designs’, J. Combin. Theory Ser. A 26 (1979), 119.
[24] Ho, T., Médard, M., Koetter, R., Karger, D., Effros, M., Shi, J. and Leong, B., ‘A random linear network coding approach to multicast’, IEEE Trans. Inform. Theory 52 (2006), 44134430.
[25] Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).
[26] Kantor, W. M., ‘Linear groups containing a Singer cycle’, J. Algebra 62 (1980), 232234.
[27] Kaski, P. and Östergård, P. R. J., Classification Algorithms for Codes and Designs (Springer, Berlin, 2006).
[28] Kaski, P. and Pottonen, O., ‘Libexact user guide, Version 1.0’, HIIT Technical Reports 2008-1, Helsinki Institute for Information Technology, 2008.
[29] Keevash, P., ‘The existence of designs’, Preprint 2014, arXiv:1401.3665.
[30] Kiermaier, M. and Laue, R., ‘Derived and residual subspace designs’, Adv. Math. Commun. 9 (2015), 105115.
[31] Kirkman, T. P., ‘On a problem in combinations’, Cambridge Dublin Math. J. II (1847), 191204.
[32] Knuth, D. E., ‘Dancing links’, inMillennial Perspectives in Computer Science (eds. Davies, J., Roscoe, B. and Woodcock, J.) (Palgrave Macmillan, Basingstoke, 2000), 187214.
[33] Koelink, E. and van Assche, W., ‘Leonhard Euler and a q-analogue of the logarithm’, Proc. Amer. Math. Soc. 137 (2009), 16631676.
[34] Koetter, R. and Kschischang, F. R., ‘Coding for errors and erasures in random network coding’, IEEE Trans. Inform. Theory 54 (2008), 35793591.
[35] Kohnert, A. and Kurz, S., ‘Construction of large constant dimension codes with a prescribed minimum distance’, inMathematical Methods in Computer Science, (eds. Calmet, J., Geiselmann, W. and Müller-Quade, J.) Lecture Notes in Computer Science, 5393 (Springer, Berlin, 2008), 3142.
[36] Kramer, E. and Mesner, D., ‘ t-designs on hypergraphs’, Discrete Math. 15 (1976), 263296.
[37] Laue, R., ‘Eine konstruktive Version des Lemmas von Burnside’, Bayreuther Math. Schr. 28 (1989), 111125.
[38] van Lint, J. H. and Wilson, R. M., A Course in Combinatorics, 2nd edn (Cambridge University Press, Cambridge, 2001).
[39] Metsch, K., ‘Bose-Burton type theorems for finite projective, affine and polar spaces’, inSurveys in Combinatorics (eds. Lamb, J. D. and Preece, D. A.) London Math. Soc. Lecture Note, 267 (Cambridge University Press, Cambridge, 1999), 137166.
[40] Miyakawa, M., Munemasa, A. and Yoshiara, S., ‘On a class of small 2-designs over GF(q)’, J. Combin. Des. 3 (1995), 6177.
[41] Plücker, J., System der analytischen Geometrie: auf neue Betrachtungsweisen gegründet, und insbesondere eine ausführliche Theorie der Curven dritter Ordnung enthaltend, (Duncker und Humblot, Berlin, 1835).
[42] Ray-Chaudhuri, D. K. and Singhi, N. M., ‘ q-analogues of t-designs and their existence’, Linear Algebra Appl. 114/115 (1989), 5768.
[43] Schmalz, B., ‘The t-designs with prescribed automorphism group, new simple 6-designs’, J. Combin. Des. 1 (1993), 125170.
[44] Schwartz, M. and Etzion, T., ‘Codes and anticodes in the Grassmann graph’, J. Combin. Theory Ser. A 97 (2002), 2742.
[45] Steiner, J., ‘Combinatorische Aufgabe’, J. reine angew. Math. 45 (1853), 181182.
[46] Suzuki, H., ‘2-designs over GF(2 m )’, Graphs Combin. 6 (1990), 293296.
[47] Suzuki, H., ‘2-designs over GF(q)’, Graphs Combin. 8 (1992), 381389.
[48] Teirlinck, L., ‘Non-trivial t-designs without repeated blocks exist for all t ’, Discrete Math. 65 (1987), 301311.
[49] Thomas, S., ‘Designs over finite fields’, Geom. Dedicata 21 (1987), 237242.
[50] Thomas, S., ‘Designs and partial geometries over finite fields’, Geom. Dedicata 63 (1996), 247253.
[51] Tits, J., ‘Sur les analogues algébriques des groupes semi-simples complexes’, inColloque d’Algébre Supèrieure, tenu á Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Établissements Ceuterick (Louvain, Paris, Librairie Gauthier-Villars, 1957), 261289.
[52] Wang, J., ‘Quotient sets and subset-subspace analogy’, Adv. Appl. Math. 23 (1999), 333339.
[53] Wilson, R. M., ‘Cyclotomy and difference families in elementary abelian groups’, J. Number Theory 4 (1972), 1747.
[54] Yeung, R. W., Information Theory and Network Coding (Springer, Berlin, 2008).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Forum of Mathematics, Pi
  • ISSN: -
  • EISSN: 2050-5086
  • URL: /core/journals/forum-of-mathematics-pi
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification

Type Description Title
Supplementary materials

Braun supplementary material
Braun supplementary material

 Unknown (43 KB)
43 KB


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed