Startseite Mathematik On cutting blocking sets and their codes
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On cutting blocking sets and their codes

  • Daniele Bartoli , Antonio Cossidente , Giuseppe Marino und Francesco Pavese ORCID logo EMAIL logo
Veröffentlicht/Copyright: 6. Januar 2022
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 34 Heft 2

Abstract

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 Π∩𝒳 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.

Keywords: Cutting blocking sets; minimal codes; saturating sets; covering codes
MSC 2010: 51E21; 94B05; 94B25; 51E20

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).

References

[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.2020104Suche in Google Scholar

[2] G. N. Alfarano, M. Borello, A. Neri and A. Ravagnani, Three combinatorial perspectives on minimal codes, preprint (2020), https://arxiv.org/abs/2010.16339. 10.1137/21M1391493Suche 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/1103408578Suche 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.2020.101723Suche in Google Scholar

[5] D. Bartoli, G. Micheli, G. Zini and F. Zullo, r-fat linearized polynomials over finite fields, preprint (2020), https://arxiv.org/abs/2012.15357. 10.1016/j.jcta.2022.105609Suche 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/s10623-014-9971-3Suche 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.2015.12.018Suche 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-2017-0032Suche 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/s10801-019-00930-6Suche 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.1996.0125Suche in Google Scholar

[11] A. A. Bruen, Intersection of Baer subgeometries, Arch. Math. (Basel) 39 (1982), no. 3, 285–288. 10.1007/BF01899537Suche in Google Scholar

[12] G. Cohen, I. Honkala, S. Litsyn and A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. Suche 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.2011.5.119Suche in Google Scholar

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

[15] U. Dempwolff, A note on the Figueroa planes, Arch. Math. (Basel) 43 (1984), no. 3, 285–288. 10.1007/BF01247576Suche 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-ySuche 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/s10623-010-9371-2Suche 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-xSuche 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. 10.37236/1632Suche 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. 10.37236/4149Suche 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:1024196207146Suche 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/BF00183019Suche 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/s10623-014-9941-9Suche in Google Scholar

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

[25] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2nd ed., Oxford Math. Monogr., The Clarendon, New York, 1998. 10.1093/oso/9780198502951.001.0001Suche in Google Scholar

[26] J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Springer Monogr. Math., Springer, London, 2016. 10.1007/978-1-4471-6790-7Suche 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/s10623-010-9393-9Suche 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/s10801-013-0468-3Suche 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.2004.029Suche 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.2021.3070377Suche 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/139Suche 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

Artikel in diesem Heft

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
https://doi.org/10.1515/forum-2020-0338
Schlagwörter für diesen Artikel
Cutting blocking sets; minimal codes; saturating sets; covering codes

Artikel in diesem Heft

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
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