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Combinatorica

, Volume 37, Issue 6, pp 1073–1095 | Cite as

Subspace Codes in PG(2N − 1, Q)

  • Antonio Cossidente
  • Francesco Pavese
Original Paper

Abstract

An (r,M,2δ;k) q constant-dimension subspace code, δ > 1, is a collection C of (k − 1)-dimensional projective subspaces of PG(r − 1,q) such that every (kδ)-dimensional projective subspace of PG(r − 1,q) is contained in at most one member of C. Constant-dimension subspace codes gained recently lot of interest due to the work by Koetter and Kschischang [20], where they presented an application of such codes for error-correction in random network coding. Here a (2n,M,4;n) q constant-dimension subspace code is constructed, for every n ≥ 4. The size of our codes is considerably larger than all known constructions so far, whenever n > 4. When n = 4 a further improvement is provided by constructing an (8,M,4;4) q constant-dimension subspace code, with M = q12 + q2(q2 + 1)2(q2 + q + 1) + 1.

Mathematics Subject Classification (2000)

51E20 05B25 94B27 94B60 94B65 

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References

  1. [1]
    A. Beutelspacher: Partial spreads in finite projective spaces and partial designs, Math. Z. 145 (1975), 211–229.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    J. M. N. Brown: Partitioning the complement of a simplex in PG(e,q d+1) into copies of PG(d,q), J. Geom. 33 (1988), 11–16.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Cossidente and F. Pavese: On subspace codes, Des. Codes Cryptogr. DOI 10.1007/s10623-014-0018-6.Google Scholar
  4. [4]
    A. Cossidente, G. Marino and F. Pavese: Non-linear maximum rank distance codes, Des. Codes Cryptogr. DOI 10.1007/s10623-015-0108-0.Google Scholar
  5. [5]
    P. Delsarte: Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A 25 (1978), 226–241.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    R. H. Dye: Spreads and classes of maximal subgroups of GL(n,q), SL(n,q), PGL(n,q) and PSL(n,q), Ann. Mat. Pura Appl. 158 (1991), 33–50.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    T. Etzion and N. Silberstein: Codes and Designs Related to Lifted MRD Codes, IEEE Trans. Inform. Theory 59 (2013), 1004–1017.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. Etzion and N. Silberstein: Error-correcting codes in projective spaces via rankmetric codes and Ferrers diagrams, IEEE Trans. Inform. Theory 55 (2009), 2909–2919.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    T. Etzion and A. Vardy: Error-correcting codes in projective space, IEEE Trans. Inform. Theory 57 (2011), 1165–1173.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    E. M. Gabidulin: Theory of codes with maximum rank distance, Problems of Information Transmission 21 (1985), 1–12.MathSciNetzbMATHGoogle Scholar
  11. [11]
    M. Gadouleau and Z. Yan: Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory 56 (2010), 3207–3216.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Gill: Polar spaces and embeddings of classical groups, New Zealand J. Math. 36 (2007), 175–184.MathSciNetzbMATHGoogle Scholar
  13. [13]
    J. W. P. Hirschfeld: Finite projective spaces of three dimensions, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1985.zbMATHGoogle Scholar
  14. [14]
    J. W. P. Hirschfeld and J. A. Thas: General Galois Geometries, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991.zbMATHGoogle Scholar
  15. [15]
    T. Honold, M. Kiermaier and S. Kurz: Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4, Topics in finite fields 157–176, Contemp. Math. 632, Amer. Math. Soc., Providence, RI, 2015.Google Scholar
  16. [16]
    B. Huppert: Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin-New York, 1967.Google Scholar
  17. [17]
    A. Khaleghi, D. Silva and F. R. Kschischang: Subspace codes, Cryptography and coding 1–21, Lecture Notes in Comput. Sci., Springer, Berlin, 2009.Google Scholar
  18. [18]
    P. Kleidman and M. Liebeck: The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990.CrossRefzbMATHGoogle Scholar
  19. [19]
    A. Klein, K. Metsch and L. Storme: Small maximal partial spreads in classical finite polar spaces, Adv. Geom. 10 (2010), 379–402.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    R. Koetter and F. R. Kschischang: Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory 54 (2008), 3579–3591.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Lavrauw and G. Van de Voorde: Field reduction and linear sets in finite geometry, Topics in finite fields, 271–293, Contemp. Math., 632, Amer. Math. Soc., Providence, RI, 2015.Google Scholar
  22. [22]
    G. Lunardon, G. Marino, O. Polverino and R. Trombetti: Maximum scattered linear sets of pseudoregulus type and the Segre Variety Sn;n, J. Algebr. Comb. 39 (2014), 807–831.CrossRefzbMATHGoogle Scholar
  23. [23]
    R. M. Roth: Maximum-rank array codes and their application to crisscross error correction, IEEE Trans. Inform. Theory 37 (1991), 328–336.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    B. Segre: Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl. 64 (1964), 1–76.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    D. Silva, F. R. Kschischang and R. Koetter: A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951–3967.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A.-L. Trautmann and J. Rosenthal: New improvements on the echelon-Ferrers construction, in: Proc. of Int. Symp. on Math. Theory of Networks and Systems, 405–408, 2010.Google Scholar
  27. [27]
    S.-T. Xia and F.-W. Fu: Johnson type bounds on constant dimension codes, Des. Codes Cryptogr. 50 (2009), 163–172.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Dipartimento di Matematica, Informatica ed EconomiaUniversità della Basilicata, Contrada Macchia RomanaPotenzaItaly
  2. 2.Dipartimento di Meccanica, Matematica e ManagementPolitecnico di BariBariItaly

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