Abstract
We consider the structure of the point-line incidence matrix of the projective space \(\mathrm {PG}(3,q)\) connected with orbits of points and lines under the stabilizer group of the twisted cubic. Structures of submatrices with incidences between a union of line orbits and an orbit of points are investigated. For the unions consisting of two or three line orbits, the original submatrices are split into new ones, in which the incidences are also considered. For each submatrix (apart from the ones corresponding to a special type of lines), the numbers of lines through every point and of points lying on every line are obtained. This corresponds to the numbers of ones in columns and rows of the submatrices.
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References
Ball S., Lavrauw M.: Arcs in finite projective spaces. EMS Surv. Math. Sci. 6(1/2), 133–172 (2019).
Barg A., Zemor G.: Distances properties of expander codes. IEEE Trans. Inf. Theory 52(1), 78–90 (2006).
Bartoli D., Davydov A.A., Marcugini S., Pambianco F.: On planes through points off the twisted cubic in PG(3, q) and multiple covering codes. Finite Fields Appl. 67, 101710 (2020).
Blokhuis A., Pellikaan R., Szönyi T.: The extended coset leader weight enumerator of a twisted cubic code, arXiv:2103.16904 [math.CO] (2021).
Bonoli G., Polverino O.: The twisted cubic in PG(3, q) and translation spreads in \(H(q)\). Discret. Math. 296, 129–142 (2005).
Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235–265 (1997).
Bruen A.A., Hirschfeld J.W.P.: Applications of line geometry over finite fields I: The twisted cubic. Geom. Dedicata 6, 495–509 (1977).
Cardinali I., Lunardon G., Polverino O., Trombetti R.: Spreads in \(H(q)\) and 1-systems of \(Q(6, q)\). Eur. J. Combin. 23, 367–376 (2002).
Casse L.R.A., Glynn D.G.: The solution to Beniamino Segre’s problem \(I_{r, q}\), \(r = 3\), \(q = 2^h\). Geom. Dedicata 13, 157–163 (1982).
Casse L.R.A., Glynn D.G.: On the uniqueness of \((q + 1)_{4}\)-arcs of PG(4, q), \( q= 2^h\), \(h\ge 3\). Discret. Math. 48(2–3), 173–186 (1984).
Cossidente A., Hirschfeld J.W.P., Storme L.: Applications of line geometry, III: The quadric Veronesean and the chords of a twisted cubic. Austral. J. Combin. 16, 99–111 (1997).
Davydov A.A., Giulietti M., Marcugini S., Pambianco F.: Some combinatorial aspects of constructing bipartite-graph codes. Graphs Combin. 29(2), 187–212 (2013).
Davydov A.A., Faina G., Giulietti M., Marcugini S., Pambianco F.: On constructions and parameters of symmetric configurations \(v_k\). Des. Codes Cryptogr. 80(1), 125–147 (2016).
Davydov A.A., Marcugini S., Pambianco F.: Twisted cubic and orbits of lines in PG(3,q), arXiv:2103.12655 [math.CO] (2021).
Davydov A.A., Marcugini S., Pambianco F.: Twisted cubic and plane-line incidence matrix in PG(3,q), arXiv:2103.11248 [math.CO] (2021).
Davydov A.A., Marcugini S., Pambianco F.: On cosets weight distributions of the doubly-extended Reed-Solomon codes of codimension 4. IEEE Trans. Inform. Theory (2021). https://doi.org/10.1109/TIT.2021.3089129.
Giulietti M., Vincenti R.: Three-level secret sharing schemes from the twisted cubic. Discret. Math. 310, 3236–3240 (2010).
Gropp H.: Configurations. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 353–355. Chapman and Hall/CRC, New York (2006).
Günay G., Lavrauw M.: On pencils of cubics on the projective line over finite fields of characteristic \( > 3\), arXiv:2104.04756 [math.CO][math.AG] (2021).
Hirschfeld J.W.P.: Finite Projective Spaces of Three Dimensions. Oxford University Press, Oxford (1985).
Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Oxford University Press, Oxford (1999).
Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite projective spaces: Update 2001. In: Blokhuis A., Hirschfeld J.W.P., Jungnickel D., Thas J.A. (eds.) Finite Geometries (Proc. 4th Isle of Thorns Conf., July 16-21, 2000), Dev. Math.), vol. 3, pp. 201–246. Kluwer, Dordrecht (2001).
Hirschfeld J.W.P., Thas J.A.: Open problems in finite projective spaces. Finite Fields Appl. 32, 44–81 (2015).
Høholdt T., Justesen J.: Graph codes with Reed-Solomon component codes. Proc. Int. Symp. Inf. Theory 2006, ISIT 2006, pp. 2022–2026. IEEE, Seattle (2006).
Lidl R., Niederreiter H.: Introduction to Finite Fields and their Applications, 2nd edn. Cambridge University Press, Cambridge (1994).
Lunardon G., Polverino O.: On the Twisted Cubic of PG(3, q). J. Algebr. Combin. 18, 255–262 (2003).
Mathon R., Rosa A.: \(2\)-\((v, k,\lambda )\) Designs of Small Order. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 25–58. Chapman and Hall/CRC, New York (2006).
Zannetti M., Zuanni F.: Note on three-character \((q + 1)\)-sets in PG(3, q). Austral. J. Combin. 47, 37–40 (2010).
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Communicated by J. W. P. Hirschfeld.
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The research of S. Marcugini, and F. Pambianco was supported in part by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INDAM) (Contract No. U-UFMBAZ-2019-000160, 11.02.2019) and by University of Perugia (Project No. 98751: Strutture Geometriche, Combinatoria e loro Applicazioni, Base Research Fund 2017-2019)
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Davydov, A.A., Marcugini, S. & Pambianco, F. Twisted cubic and point-line incidence matrix in \(\mathrm {PG}(3,q)\). Des. Codes Cryptogr. 89, 2211–2233 (2021). https://doi.org/10.1007/s10623-021-00911-6
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DOI: https://doi.org/10.1007/s10623-021-00911-6