Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter January 6, 2022

On cutting blocking sets and their codes

  • Daniele Bartoli , Antonio Cossidente , Giuseppe Marino and Francesco Pavese ORCID logo EMAIL logo
From the journal Forum Mathematicum


Let PG(r,q) be the r-dimensional projective space over the finite field GF(q). A set 𝒳 of points of PG(r,q) is a cutting blocking set if for each hyperplane Π of PG(r,q) the set Π𝒳 spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG(3,q3) of size 3(q+1)(q2+1) as a union of three pairwise disjoint q-order subgeometries, and a cutting blocking set of PG(5,q) of size 7(q+1) from seven lines of a Desarguesian line spread of PG(5,q). In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q-ary linear code having dimension 4 and 6.

Communicated by Manfred Droste

Funding statement: This work was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA–INdAM).


[1] G. N. Alfarano, M. Borello and A. Neri, A geometric characterization of minimal codes and their asymptotic performance, Adv. Math. Commun. (2020), 10.3934/amc.2020104. 10.3934/amc.2020104Search in Google Scholar

[2] G. N. Alfarano, M. Borello, A. Neri and A. Ravagnani, Three combinatorial perspectives on minimal codes, preprint (2020), 10.1137/21M1391493Search in Google Scholar

[3] R. D. Baker, J. M. N. Brown, G. L. Ebert and J. C. Fisher, Projective bundles, Bull. Belg. Math. Soc. Simon Stevin 1 (1994), 329–336. 10.36045/bbms/1103408578Search in Google Scholar

[4] D. Bartoli, G. Kiss, S. Marcugini and F. Pambianco, Resolving sets for higher dimensional projective spaces, Finite Fields Appl. 67 (2020), Article ID 101723. 10.1016/j.ffa.2020.101723Search in Google Scholar

[5] D. Bartoli, G. Micheli, G. Zini and F. Zullo, r-fat linearized polynomials over finite fields, preprint (2020), 10.1016/j.jcta.2022.105609Search in Google Scholar

[6] S. G. Barwick and W.-A. Jackson, An investigation of the tangent splash of a subplane of PG(2,q3), Des. Codes Cryptogr. 76 (2015), no. 3, 451–468. 10.1007/s10623-014-9971-3Search in Google Scholar

[7] S. G. Barwick and W.-A. Jackson, Exterior splashes and linear sets of rank 3, Discrete Math. 339 (2016), no. 5, 1613–1623. 10.1016/j.disc.2015.12.018Search in Google Scholar

[8] S. G. Barwick and W.-A. Jackson, The exterior splash in PG(6,q): Carrier conics, Adv. Geom. 17 (2017), no. 4, 407–422. 10.1515/advgeom-2017-0032Search in Google Scholar

[9] M. Bonini and M. Borello, Minimal linear codes arising from blocking sets, J. Algebraic Combin. 53 (2021), no. 2, 327–341. 10.1007/s10801-019-00930-6Search in Google Scholar

[10] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265. 10.1006/jsco.1996.0125Search in Google Scholar

[11] A. A. Bruen, Intersection of Baer subgeometries, Arch. Math. (Basel) 39 (1982), no. 3, 285–288. 10.1007/BF01899537Search in Google Scholar

[12] G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. Search in Google Scholar

[13] A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Linear nonbinary covering codes and saturating sets in projective spaces, Adv. Math. Commun. 5 (2011), no. 1, 119–147. 10.3934/amc.2011.5.119Search in Google Scholar

[14] A. A. Davydov, S. Marcugini and F. Pambianco, New covering codes of radius R, codimension tR and tR+R2, and saturating sets in projective spaces, Des. Codes Cryptogr. 87 (2019), no. 12, 2771–2792. 10.1007/s10623-019-00649-2Search in Google Scholar

[15] U. Dempwolff, A note on the Figueroa planes, Arch. Math. (Basel) 43 (1984), no. 3, 285–288. 10.1007/BF01247576Search in Google Scholar

[16] L. Denaux, Constructing saturating sets in projective spaces using subgeometries, Des. Codes Cryptogr. (2021), 10.1007/s10623-021-00951-y. 10.1007/s10623-021-00951-ySearch in Google Scholar

[17] G. Donati and N. Durante, On the intersection of a Hermitian curve with a conic, Des. Codes Cryptogr. 57 (2010), no. 3, 347–360. 10.1007/s10623-010-9371-2Search in Google Scholar

[18] G. Donati and N. Durante, Scattered linear sets generated by collineations between pencils of lines, J. Algebraic Combin. 40 (2014), no. 4, 1121–1134. 10.1007/s10801-014-0521-xSearch in Google Scholar

[19] K. Drudge, On the orbits of Singer groups and their subgroups, Electron. J. Combin. 9 (2002), no. 1, Research Paper 15. 10.37236/1632Search in Google Scholar

[20] S. L. Fancsali and P. Sziklai, Lines in higgledy-piggledy arrangement, Electron. J. Combin. 21 (2014), no. 2, Paper No. 2.56. 10.37236/4149Search in Google Scholar

[21] S. Ferret and L. Storme, Results on maximal partial spreads in PG(3,p3) and on related minihypers, Des. Codes Cryptogr. 29 (2003), 105–122. 10.1023/A:1024196207146Search in Google Scholar

[22] D. G. Glynn, On a set of lines of PG(3,q) corresponding to a maximal cap contained in the Klein quadric of PG(5,q), Geom. Dedicata 26 (1988), no. 3, 273–280. 10.1007/BF00183019Search in Google Scholar

[23] T. Héger, B. Patkós and M. Takáts, Search problems in vector spaces, Des. Codes Cryptogr. 76 (2015), no. 2, 207–216. 10.1007/s10623-014-9941-9Search in Google Scholar

[24] J. W. P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford Math. Monogr., The Clarendon, New York, 1985. Search in Google Scholar

[25] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2nd ed., Oxford Math. Monogr., The Clarendon, New York, 1998. Search in Google Scholar

[26] J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Springer Monogr. Math., Springer, London, 2016. 10.1007/978-1-4471-6790-7Search in Google Scholar

[27] M. Lavrauw and G. Van de Voorde, On linear sets on a projective line, Des. Codes Cryptogr. 56 (2010), no. 2–3, 89–104. 10.1007/s10623-010-9393-9Search in Google Scholar

[28] G. Lunardon, G. Marino, O. Polverino and R. Trombetti, Maximum scattered linear sets of pseudoregulus type and the Segre variety 𝒮n,n, J. Algebraic Combin. 39 (2014), no. 4, 807–831. 10.1007/s10801-013-0468-3Search in Google Scholar

[29] G. Lunardon and O. Polverino, Translation ovoids of orthogonal polar spaces, Forum Math. 16 (2004), no. 5, 663–669. 10.1515/form.2004.029Search in Google Scholar

[30] C. Tang, Y. Qiu, Q. Liao and Z. Zhou, Full characterization of minimal linear codes as cutting blocking sets, IEEE Trans. Inform. Theory 67 (2021), no. 6, 3690–3700. 10.1109/TIT.2021.3070377Search in Google Scholar

[31] M. Tsfasman, S. Vlăduţ and D. Nogin, Algebraic Geometric Codes: Basic Notions, Math. Surveys Monogr. 139, American Mathematical Society, Providence, 2007. 10.1090/surv/139Search in Google Scholar

Received: 2020-12-01
Published Online: 2022-01-06
Published in Print: 2022-03-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 5.12.2023 from
Scroll to top button