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Scattered subspaces and related codes

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Abstract

After a seminal paper by Shekeey (Adv Math Commun 10(3):475-488, 2016), a connection between maximum h-scattered \({{\mathbb {F}}}_{q}\)-subspaces of \(V(r,q^n)\) and maximum rank distance (MRD) codes has been established in the extremal cases \(h=1\) and \(h=r-1\). In this paper, we propose a connection for any \(h\in \{1,\ldots ,r-1\}\), extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. We show that, up to equivalence, MRD codes having the same parameters as the ones in our connection come from an h-scattered subspace. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum h-scattered subspaces.

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Notes

  1. Recall that \(\langle f\rangle _{{\mathcal {F}}_n}=\{f\circ \omega _{\alpha }:\alpha \in {\mathbb {F}}_{q^n}\}\).

References

  1. Ball S., Blokhuis A., Lavrauw M.: Linear \((q+1)\)-fold blocking sets in \(\text{ PG }(2, q^4)\). Finite Fields Appl. 6(4), 294–301 (2000).

    Article  MathSciNet  Google Scholar 

  2. Bartoli D., Csajbók B., Marino G., Trombetti R.: Evasive subspaces, arXiv:2005.08401.

  3. Bartoli D., Giulietti M., Marino G., Polverino O.: Maximum scattered linear sets and complete caps in Galois spaces. Combinatorica 38(2), 255–278 (2018).

    Article  MathSciNet  Google Scholar 

  4. Bartoli D., Giulietti M., Zini G.: Exceptional scatteredness in odd degree. In preparation.

  5. Bartoli D., Montanucci M.: On the classification of exceptional scattered polynomials. J. Comb. Theory Ser. A 179, 105386 (2021).

    Article  MathSciNet  Google Scholar 

  6. Bartoli D., Zhou Y.: Exceptional scattered polynomials. J. Algebra 509, 507–534 (2018).

    Article  MathSciNet  Google Scholar 

  7. Bartoli D., Zhou Y.: Asymptotics of Moore exponent sets. J. Comb. Theory Ser. A 175, 105281 (2020).

    Article  MathSciNet  Google Scholar 

  8. Bartoli D., Zini G., Zullo F.: Linear maximum rank distance codes of exceptional type. In preparation.

  9. Blokhuis A., Lavrauw M.: Scattered spaces with respect to a spread in \(\rm PG(n, q)\). Geom. Dedicata 81, 231–243 (2000).

    Article  MathSciNet  Google Scholar 

  10. Byrne E., Ravagnani A.: Covering radius of matrix codes endowed with the rank metric. SIAM J. Discret. Math. 31, 927–944 (2017).

    Article  MathSciNet  Google Scholar 

  11. Csajbók B., Marino G., Polverino O., Zullo F.: Generalising the scattered property of subspaces. Combinatorica (2021). https://doi.org/10.1007/s00493-020-4347-y.

    Article  MathSciNet  Google Scholar 

  12. Csajbók B., Marino G., Polverino O., Zullo F.: Maximum scattered linear sets and MRD-codes. J. Algebraic Comb. 46, 1–15 (2017).

    Article  MathSciNet  Google Scholar 

  13. Csajbók B., Siciliano A.: Puncturing maximum rank distance codes. J. Algebraic Comb. 49, 507–534 (2019).

    Article  MathSciNet  Google Scholar 

  14. Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25, 226–241 (1978).

    Article  MathSciNet  Google Scholar 

  15. Ferraguti A., Micheli G.: Exceptional scatteredness in prime degree. J. Algebra 565, 691–701 (2021).

    Article  MathSciNet  Google Scholar 

  16. Gabidulin E.: Theory of codes with maximum rank distance. Problems Inf. Trans. 21(3), 3–16 (1985).

    MathSciNet  MATH  Google Scholar 

  17. Giuzzi L., Zullo F.: Identifiers for MRD-codes. Linear Algebra Appl. 575, 66–86 (2019).

    Article  MathSciNet  Google Scholar 

  18. Hiss G.: Finite groups of Lie type and their representations. In: Campbell C.M., Quick M.R., Robertson E.F., Roney-Dougal C.M., Smith G.C., Traustason G. (eds.) Groups St Andrews 2009 in Bath, pp. 1–40. Cambridge University Press, Cambridge (2011).

    Google Scholar 

  19. Huppert B.: Endliche Gruppen, vol. 1. Springer, Berlin (1967).

    Book  Google Scholar 

  20. Kshevetskiy A., Gabidulin E.: The new construction of rank codes. In: International Symposium on Information Theory, 2005. ISIT 2005. Proceedings, pp. 2105–2108 (2005).

  21. Lavrauw M.: Scattered spaces in Galois geometry. Contemp. Dev. Finite Fields Appl. 195–216 (2016).

  22. Lavrauw M., Van de Voorde G.: Field reduction and linear sets in finite geometry. In: Topics in Finite Fields, vol. 623, pp. 271–293. AMS Contemporary Math. American Mathematical Society, Providence (2015).

  23. Liebhold D., Nebe G.: Automorphism groups of Gabidulin-like codes. Arch. Math. 107(4), 355–366 (2016).

    Article  MathSciNet  Google Scholar 

  24. Lunardon G.: MRD-codes and linear sets. J. Comb. Theory Ser. A 149, 1–20 (2017).

    Article  MathSciNet  Google Scholar 

  25. Lunardon G.: Normal spreads. Geom. Dedicata 75, 245–261 (1999).

    Article  MathSciNet  Google Scholar 

  26. Lunardon G., Polito P., Polverino O.: A geometric characterisation of linear k-blocking sets. J. Geom. 74(1–2), 120–122 (2002).

    Article  MathSciNet  Google Scholar 

  27. Lunardon G., Polverino O.: Translation ovoids of orthogonal polar spaces. Forum Math. 16, 663–669 (2004).

    Article  MathSciNet  Google Scholar 

  28. Lunardon G., Trombetti R., Zhou Y.: Generalized twisted Gabidulin codes. J. Comb. Theory Ser. A 159, 79–106 (2018).

    Article  MathSciNet  Google Scholar 

  29. Lunardon G., Trombetti R., Zhou Y.: On kernels and nuclei of rank metric codes. J. Algebraic Comb. 46, 313–340 (2017).

    Article  MathSciNet  Google Scholar 

  30. Napolitano V., Polverino O., Zini G., Zullo F.: Linear sets from projection of Desarguesian spreads. Finite Fields Appl. 71, 1–19 (2021).

    Article  MathSciNet  Google Scholar 

  31. Napolitano V., Zullo F.: Codes with few weights arising from linear sets. Adv. Math. Commun. (2020). https://doi.org/10.3934/amc.2020129.

    Article  Google Scholar 

  32. Neri A., Puchinger S., Horlemann-Trautmann A.: Equivalence and characterizations of linear rank-metric codes based on invariants. Linear Algebra Appl. 603, 418–469 (2020).

    Article  MathSciNet  Google Scholar 

  33. Polverino O.: Linear sets in finite projective spaces. Discret. Math. 310(22), 3096–3107 (2010).

    Article  MathSciNet  Google Scholar 

  34. Polverino O., Zullo F.: Connections between scattered linear sets and MRD-codes. Bull. ICA 89, 46–74 (2020).

    MathSciNet  MATH  Google Scholar 

  35. Randrianarisoa T.H.: A geometric approach to rank metric codes and a classification of constant weight codes. Des. Codes Cryptogr. 80(1), 750 (2020). https://doi.org/10.1007/s10623-020-00750-x.

    Article  MathSciNet  MATH  Google Scholar 

  36. Ravagnani A.: Rank-metric codes and their duality theory. Des. Codes Cryptogr. 80(1), 197–216 (2016).

    Article  MathSciNet  Google Scholar 

  37. Segre B.: Teoria di Galois, fibrazioni proiettive e geometrie non Desarguesiane. Ann. Mat. Pura Appl. 64, 1–76 (1964).

    Article  MathSciNet  Google Scholar 

  38. Sheekey J.: A new family of linear maximum rank distance codes. Adv. Math. Commun. 10(3), 475–488 (2016).

    Article  MathSciNet  Google Scholar 

  39. Sheekey J.: New semifields and new MRD codes from skew polynomial rings. J. Lond. Math. Soc. 101(1), 432–456 (2020).

    Article  MathSciNet  Google Scholar 

  40. Sheekey J.: MRD codes: constructions and connections. Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications, Radon Series on Computational and Applied Mathematics 23, K.-U. Schmidt and A. Winterhof (eds.), De Gruyter (2019).

  41. Sheekey J., Van de Voorde G.: Rank-metric codes, linear sets and their duality. Des. Codes Cryptogr. (2019). https://doi.org/10.1007/s10623-019-00703-z.

    Article  MATH  Google Scholar 

  42. Trombetti R., Zhou Y.: Nuclei and automorphism groups of generalized twisted Gabidulin codes. Linear Algebra Appl. 575, 1–26 (2019).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, 81100, Caserta, Italy

    Giovanni Zini & Ferdinando Zullo

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  1. Giovanni Zini
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  2. Ferdinando Zullo
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Correspondence to Ferdinando Zullo.

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Communicated by G. Lunardon.

Dedicated to the memory of Elisa Montanucci. We join in her family’s pain.

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Giovanni Zini is funded by the project “Attrazione e Mobilità dei Ricercatori” Italian PON Programme (PON-AIM 2018 num. AIM1878214-2). The research was supported by the project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania “Luigi Vanvitelli”, and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM).

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Zini, G., Zullo, F. Scattered subspaces and related codes. Des. Codes Cryptogr. 89, 1853–1873 (2021). https://doi.org/10.1007/s10623-021-00891-7

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