Advertisement

Designs, Codes and Cryptography

, Volume 78, Issue 1, pp 311–350 | Cite as

Galois geometries and coding theory

  • T. Etzion
  • L. Storme
Article

Abstract

Galois geometries and coding theory are two research areas which have been interacting with each other for many decades. From the early examples linking linear MDS codes with arcs in finite projective spaces, linear codes meeting the Griesmer bound with minihypers, covering radius with saturating sets, links have evolved to functional codes, generalized projective Reed–Muller codes, and even further to LDPC codes, random network codes, and distributed storage. This article reviews briefly the known links, and then focuses on new links and new directions. We present new results and open problems to stimulate the research on Galois geometries, coding theory, and on their continuously developing and increasing interactions.

Keywords

Galois geometries Coding theory Network coding  Designs and codes over vector spaces 

Mathematics Subject Classification

94B25 05B40 51E10 

Notes

Acknowledgments

This research was supported in part by the Israeli Science Foundation (ISF), Jerusalem, Israel, under Grant 10/12.

References

  1. 1.
    Ahlswede E., Cai N., Li S.-Y.R., Yeung R.W.: Network information flow. IEEE Trans. Inf. Theory 46, 1204–1216 (2000).Google Scholar
  2. 2.
    Ahlswede R., Aydinian H.K., Khachatrian L.H.: On perfect codes and related concepts. Des. Codes Cryptogr. 22, 221–237 (2001).Google Scholar
  3. 3.
    Bachoc C., Passuello A., Vallentin F.: Bounds for projective codes from semidefinite programming. Adv. Math. Commun. 7, 127–145 (2013).Google Scholar
  4. 4.
    Baker R.D.: Partitioning the planes \({\rm AG}_{2m}(2)\) into 2-designs. Discret. Math. 15, 205–211 (1976).Google Scholar
  5. 5.
    Baker R.D., van Lint J.H., Wilson R.M.: On the Preparata and Goethals codes. IEEE Trans. Inf. Theory 29, 342–345 (1983).Google Scholar
  6. 6.
    Ball S.: On sets of vectors of a finite vector space in which every subset of basis size is a basis. J. Eur. Math. Soc. 14, 733–748 (2012).Google Scholar
  7. 7.
    Ball S., De Beule J.: On sets of vectors of a finite vector space in which every subset of basis size is a basis II. Des. Codes Cryptogr. 65, 5–14 (2012).Google Scholar
  8. 8.
    Barrolleta R.D., De Boeck M., Storme L., Suárez Canedo E., Vandendriessche P.: A bound for the sunflower property (preprint).Google Scholar
  9. 9.
    Bartoli D., Storme L.: On the functional codes arising from the intersections of algebraic varieties of small degree with a non-singular quadric. Adv. Math. Commun. 8, 271–280 (2014).Google Scholar
  10. 10.
    Bartoli D., De Boeck M., Fanali S., Storme L.: On the functional codes defined by quadrics and Hermitian varieties. Des. Codes Cryptogr. 71, 21–46 (2014).Google Scholar
  11. 11.
    Bartoli D., Sboui A., Storme L.: Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed–Muller codes. Adv. Math. Commun. (to appear).Google Scholar
  12. 12.
    Belov B.I., Logachev V.N., Sandimirov V.P.: Construction of a class of linear binary codes achieving the Varshamov–Griesmer bound. Probl. Inf. Transm. 10, 211–217 (1974).Google Scholar
  13. 13.
    Ben-Sasson E., Etzion T., Gabizon A., Raviv N.: Subspace Polynomials and Cyclic Subspace Codes. arXiv:1404.7739v2 (January 2015).
  14. 14.
    Berlekamp E.R.: The technology of error-correcting codes. Proc. IEEE 68, 564–593 (1980).Google Scholar
  15. 15.
    Beutelspacher A.: On parallelisms in finite projective spaces. Geom. Dedicata 3, 35–40 (1974).Google Scholar
  16. 16.
    Beutelspacher A.: Partial spreads in finite projective spaces and partial designs. Math. Z. 145, 211–229 (1975).Google Scholar
  17. 17.
    Beutelspacher A.: Parallelismen in unendlichen projektiven Räumen endlicher Dimension. Geom. Dedicata 7, 499–506 (1978).Google Scholar
  18. 18.
    Beutelspacher A.: On \(t\)-covers in finite projective spaces. J. Geom. 12, 10–16 (1979).Google Scholar
  19. 19.
    Beutelspacher A.: Partial parallelisms in finite projective spaces. Geom. Dedicata 36, 273–278 (1990).Google Scholar
  20. 20.
    Beutelspacher A., Ueberberg J.: A characteristic property of geometric \(t\)-spreads in finite projective spaces. Eur. J. Comb. 12, 277–281 (1991).Google Scholar
  21. 21.
    Beutelspacher A., Rosenbaum U.: Projective Geometry: From Foundations to Applications. Cambridge University Press, Cambridge (1998).Google Scholar
  22. 22.
    Blackburn S., Etzion T.: The asymptotic behavior of Grassmannian codes. IEEE Trans. Inf. Theory 58, 6605–6609 (2012).Google Scholar
  23. 23.
    Blokhuis A., Lovász L., Storme L., Szőnyi T.: On multiple blocking sets in Galois planes. Adv. Geom. 7, 39–53 (2007).Google Scholar
  24. 24.
    Blokhuis A., Sziklai P., Szőnyi T.: Blocking sets in projective spaces. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry, pp. 61–84. Nova Academic Publishers, New York (2011).Google Scholar
  25. 25.
    Bose R.C., Burton R.C.: A characterization of flat spaces in finite geometry and the uniqueness of the Hamming and the MacDonald codes. J. Comb. Theory 1, 96–104 (1966).Google Scholar
  26. 26.
    Braun M.: New 3-designs over the binary field. Int. Electron. J. Geom. 6, 79–87 (2013).Google Scholar
  27. 27.
    Braun M., Reichelt J.: \(q\)-Analog of packing designs. J. Comb. Des. 22, 306–321 (2014).Google Scholar
  28. 28.
    Braun M., Kerber A., Laue R.: Systematic construction of q-analogs of \(t-(v,k,\lambda )\)-designs. Des. Codes Cryptogr. 34, 55–70 (2005).Google Scholar
  29. 29.
    Braun M., Etzion T., Östergård P.R.J., Vardy A., Wassermann A.: Existence of \(q\)-Analogs of Steiner Systems. arXiv:1304.1462 (April 2013).
  30. 30.
    Braun M., Kiermaier M., Kohnert A., Laue R.: Large sets of subspace designs. arXiv:1411.7181 (November 2014).
  31. 31.
    Braun M., Kohnert A., Östergård P.R.J., Wassermann A.: Large sets of \(t\)-designs over finite fields. J. Comb. Theory Ser. A 124, 195–202 (2014).Google Scholar
  32. 32.
    Braun M., Kiermaier M., Nakić A.: On the Automorphism Group of the Binary \(q\)-Analog of the Fano Plane. arXiv:1501.07790 (January 2015).
  33. 33.
    Cafure A., Matera G.: Improved explicit estimates on the number of solutions of equations over a finite field. Finite Fields Appl. 12, 155–185 (2006).Google Scholar
  34. 34.
    Cameron P.: Generalisation of Fisher’s inequality to fields with more than one element. In: McDonough T.P., Mavron V.C. (eds.) Combinatorics. London Mathematical Society Lecture Note Series, vol. 13, pp. 9–13. Cambridge University Press, Cambridge (1974).Google Scholar
  35. 35.
    Cameron P.: Locally symmetric designs. Geom. Dedicata 3, 65–76 (1974).Google Scholar
  36. 36.
    Cohen G., Honkala I., Litysn S., Lobstein A.: Covering Codes. North-Holland Mathematical Library 54, North-Holland, Amsterdam (1997).Google Scholar
  37. 37.
    Cohn H.: Projective geometry over \(\mathbb{F}_1\) and the Gaussian binomial coefficients. Am. Math. Mon. 111, 487–495 (2004).Google Scholar
  38. 38.
    Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs. Chapman and Hall/CRC Press, Boca Raton, FL (2007).Google Scholar
  39. 39.
    Cossidente A., Pavese F.: On subspace codes. Des. Codes Cryptogr. doi: 10.1007/s10623-014-0018-6.
  40. 40.
    Davydov A.A.: Constructions and families of covering codes and saturated sets of points in projective geometry. IEEE Trans. Inf. Theory 41, 2071–2080 (1995).Google Scholar
  41. 41.
    Davydov A.A., Östergård P.R.J.: On saturating sets in small projective geometries. Eur. J. Comb. 21, 563–570 (2000).Google Scholar
  42. 42.
    De Beule J., Metsch K., Storme L.: Characterization results on arbitrary weighted minihypers and on linear codes meeting the Griesmer bound. Adv. Math. Commun. 2, 261–272 (2008).Google Scholar
  43. 43.
    De Caen D.: Extension of a theorem of Moon and Moser on complete subgraphs. Ars Comb. 16, 5–10 (1983).Google Scholar
  44. 44.
    De Caen D.: The current status of Turán’s problem on hypergraphs. In: Frankl, P., Füredi, Z., Katona, G., Miklós, D. (eds.) Extremal Problems for Finite Sets, pp. 187–197. János Bolyai Mathematical Society, Budapest (1994)Google Scholar
  45. 45.
    Delsarte P.: Association schemes and \(t\)-designs in regular semilattices. J. Comb. Theory Ser. A 20, 230–243 (1976).Google Scholar
  46. 46.
    Delsarte P.: Bilinear forms over a finite field, with applications to coding theory. J. Comb. Theory Ser. A 25, 226–241 (1978).Google Scholar
  47. 47.
    Dentice E.F., Zanella C.: Bose–Burton type theorems for finite Grassmannians. Discret. Math. 309, 363–370 (2009).Google Scholar
  48. 48.
    Deza M.: Une propriété extrémale des plans projectifs finis dans une classe de codes equidistants. Discret. Math. 6, 343–352 (1973).Google Scholar
  49. 49.
    Deza M., Frankl P.: Every large set of equidistant \((0, +1, -1)\)-vectors forms a sunflower. Combinatorica 1, 225–231 (1981).Google Scholar
  50. 50.
    Dimakis A., Godfrey P., Wu Y., Wainwright M., Ramchandran K.: Network coding for distributed storage systems. IEEE Trans. Inf. Theory 56, 4539–4551 (2010).Google Scholar
  51. 51.
    Dimakis A., Ramchandran K., Wu Y., Suh C.: A survey on network codes for distributed storage. Proc. IEEE 99, 476–489 (2011).Google Scholar
  52. 52.
    Drake D.A., Freeman J.W.: Partial t-spreads and group constructible \((s,r,\mu )\)-nets. J. Geom. 13, 210–216 (1979).Google Scholar
  53. 53.
    Edmonds J.: Edge-disjoint branchings. In: Rustin R. (ed.) Combinatorial Algorithms, pp. 91–96. Algorithmics Press, New York (1972).Google Scholar
  54. 54.
    Edoukou F.A.B., Hallez A., Rodier F., Storme L.: On the small weight codewords of the functional codes \(C_{herm}(\text{ X })\), X a non-singular Hermitian variety. Des. Codes Cryptogr. 56, 219–233 (2010).Google Scholar
  55. 55.
    Edoukou F.A.B., Hallez A., Rodier F., Storme L.: A study of intersections of quadrics having applications on the small weight codewords of the functional codes \(C_{2}(Q)\), Q a non-singular quadric. J. Pure Appl. Algebra 214, 1729–1739 (2010).Google Scholar
  56. 56.
    Eisfeld J., Metsch K.: Blocking s-dimensional subspaces by lines in PG(\(2s,q\)). Combinatorica 17, 151–162 (1997).Google Scholar
  57. 57.
    El Rouayheb S., Ramchandran K.: Fractional repetition codes for repair in distributed storage systems. In: 48-th Annual Allerton Conference on Communications, Control and Computing, pp. 1510–1517 (2010).Google Scholar
  58. 58.
    El-Zanati S., Jordon H., Seelinger G., Sissokho P., Spence L.: The maximum size of a maximal 3-spread in a finite vector space over GF(2). Des. Codes Cryptogr. 54, 101–107 (2010).Google Scholar
  59. 59.
    Etzion T.: New lower bounds for asymmetric and unidirectional codes. IEEE Trans. Inf. Theory 37, 1696–1704 (1991).Google Scholar
  60. 60.
    Etzion T.: Perfect byte-correcting codes. IEEE Trans. Inf. Theory 44, 3140–3146 (1998).Google Scholar
  61. 61.
    Etzion T.: Covering of subspaces by subspaces. Des. Codes Cryptogr. 72, 405–421 (2014).Google Scholar
  62. 62.
    Etzion T.: Partial \(k\)-Parallelisms in finite projective spaces. J. Comb. Des. 23, 101–114 (2015).Google Scholar
  63. 63.
    Etzion T.: A New Approach to Examine \(q\)-Steiner Systems. arXiv:1507.08503 (July 2015).
  64. 64.
    Etzion T., Raviv N.: Equidistant codes in the Grassmannian. Discret. Appl. Math. 186, 87–97 (2015).Google Scholar
  65. 65.
    Etzion T., Silberstein N.: Error-correcting codes in projective space via rank-metric codes and Ferrers diagrams. IEEE Trans. Inf. Theory 55, 2909–2919 (2009).Google Scholar
  66. 66.
    Etzion T., Silberstein N.: Codes and designs related to lifted MRD codes. IEEE Trans. Inf. Theory 59, 1004–1017 (2013).Google Scholar
  67. 67.
    Etzion T., Vardy A.: Error-correcting codes in projective spaces. In: International Symposium on Information Theory, pp. 871–875 (2008).Google Scholar
  68. 68.
    Etzion T., Vardy A.: Error-correcting codes in projective spaces. IEEE Trans. Inf. Theory 57, 1165–1173 (2011).Google Scholar
  69. 69.
    Etzion T., Vardy A.: On \(q\)-analogs for Steiner systems and covering designs. Adv. Math. Commun. 5, 161–176 (2011).Google Scholar
  70. 70.
    Etzion T., Vardy A.: Automorphisms of codes in the Grassmann scheme. arXiv:1210.5724 (October 2012).
  71. 71.
    Fazeli A., Lovett S., Vardy A.: Nontrivial t-designs over finite fields exist for all \(t\). J. Comb. Theory Ser. A 127, 149–160 (2014).Google Scholar
  72. 72.
    Ferret S., Storme L., Sziklai P., Weiner Zs.: A t (mod p) result on weighted multiple \((n-k)\)-blocking sets in PG(n, q). Innov. Incid. Geom. 6/7, 169–188 (2007/2008).Google Scholar
  73. 73.
    Ford Jr. L.R., Fulkerson D.R.: Maximal flow through a network. Can. J. Math. 8, 399–404 (1956).Google Scholar
  74. 74.
    Frankl P., Wilson R.M.: The Erdős–Ko–Rado theorem for vector spaces. J. Comb. Theory Ser. A 43, 228–236 (1986).Google Scholar
  75. 75.
    Fu F., Kløve T., Luo Y., Wei V.K.: On equidistant constant weight codes. Discret. Applied Math. 128, 157–164 (2003)Google Scholar
  76. 76.
    Gabidulin E.M.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21, 1–12 (1985).Google Scholar
  77. 77.
    Giulietti M.: The geometry of covering codes: small complete caps and saturating sets in Galois spaces. In: Surveys in Combinatorics 2013. London Mathematical Society Lecture Note Series, vol. 409, pp. 51–90. Cambridge University Press, Cambridge (2013).Google Scholar
  78. 78.
    Gluesing-Luerssen H., Morrison K., Troha C.: Cyclic orbit codes and stabilizer subfield. Adv. Math. Commun. 9, 177–197 (2015).Google Scholar
  79. 79.
    Goldman J.R., Rota G.-C.: On the foundations of combinatorial theory IV: finite vector spaces and Eulerian generating functions. Stud. Appl. Math. 49, 239–258 (1970).Google Scholar
  80. 80.
    Gopalan P., Hauang C., Simitci H., Yekhanin S.: On the locality of codeword symbols. IEEE Trans. Inf. Theory 58, 6925–6934 (2012).Google Scholar
  81. 81.
    Gorla E., Ravagnani A.: Subspace Codes from Ferrers Diagram. arXiv:1405.2736 (May 2014).
  82. 82.
    Gorla E., Manganiello F., Rosenthal J.: An algebraic approach for decoding spread codes. Adv. Math. Commun. 6, 443–466 (2012).Google Scholar
  83. 83.
    Griesmer J.H.: A bound for error-correcting codes. IBM J. Res. Dev. 4, 532–542 (1960).Google Scholar
  84. 84.
    Hall J.I.: Bounds for equidistant codes and partial projective planes. Discret. Math. 17, 85–94 (1977).Google Scholar
  85. 85.
    Hallez A., Storme L.: Functional codes arising from quadric intersections with Hermitian varieties. Finite Fields Appl. 16, 27–35 (2010).Google Scholar
  86. 86.
    Hamada N., Helleseth T.: A characterization of some q-ary codes \((q > (h-1)^2, h \ge 3)\) meeting the Griesmer bound. Math. Jpn. 38, 925–940 (1993).Google Scholar
  87. 87.
    Hamada N., Helleseth T.: Codes and minihypers. Optimal codes and related topics. In: Proceedings of the EuroWorkshop on Optimal Codes and Related Topics, Sunny Beach, Bulgaria, 10–16 June, pp. 79–84 (2001).Google Scholar
  88. 88.
    Hamada N., Maekawa T.: A characterization of some q-ary codes \((q>(h-1)^2, h \ge 3)\) meeting the Griesmer bound: part 2. Math. Jpn. 46, 241–252 (1997).Google Scholar
  89. 89.
    Hirschfeld J.W.P.: Projective Geometries over Finite Fields, 2nd edn. Clarendon Press, Oxford (1998).Google Scholar
  90. 90.
    Hirschfeld J.W.P., Storme L.: The packing problem in statistics, coding theory and finite projective spaces: update 2001. Developments in Mathematics. In: Blokhuis A., Hirschfeld J.W.P., Jungnickel D., Thas J.A. (eds.) Proceedings of the Fourth Isle of Thorns Conference on Finite Geometries, Chelwood Gate, 16–21 July, vol. 3, pp. 201–246. Kluwer Academic Publishers, Dordrecht (2000).Google Scholar
  91. 91.
    Hirschfeld J.W.P., Thas J.A.: General Galois geometries. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1991).Google Scholar
  92. 92.
    Ho T., Médard M., Koetter R., Karger D.R., Effros M., Shi J., Leong B.: A random linear network coding approach to multicast. IEEE Trans. Inf. Theory 52, 4413–4430 (2006).Google Scholar
  93. 93.
    Hoffman D.G., Leonard D.A., Lindner C.C., Phelps K.T., Rodger C.A., Wall J.R.: Coding Theory: The Essentials. Marcel Dekker, New York (1992).Google Scholar
  94. 94.
    Hollmann H.: Storage codes; coding rate and repair locality. In: International Conference on Computing, Networking and Communications (ICNC), pp. 830–834 (2013).Google Scholar
  95. 95.
    Hong S.J., Patel A.M.: A general class of maximal codes for computer applications. IEEE Trans. Comput. 21, 1322–1331 (1972).Google Scholar
  96. 96.
    Honold T., Kiermaier M., Kurz S.: Optimal binary subspace codes of length 6, constant dimension 3 and minimum distance 4. Contemp. Math. 632, 157–176 (2015).Google Scholar
  97. 97.
    Hsieh W.N.: Intersection theorems for systems of finite vector spaces. Discret. Math. 12, 1–16 (1975).Google Scholar
  98. 98.
    Itoh T.: A new family of 2-designs over GF(q) admitting \(\text{ SL }_m(q^\ell )\). Geom. Dedicata 69, 261–286 (1998).Google Scholar
  99. 99.
    Jaggi S., Sanders P., Chou P.A., Effros M., Egner S., Jain K., Tolhuizen L.M.G.M.: Polynomial time algorithms for multicast network code construction. IEEE Trans. Inf. Theory 51, 1973–1982 (2005).Google Scholar
  100. 100.
    Jain K., Mahdian M., Salavatipour M.R.: Packing Steiner trees. In: Proceedings of the SODA 2003, Baltimore MD, pp. 266–274 (2003).Google Scholar
  101. 101.
    Karp R.: Reducibility among combinatorial problems. In: Miller R.E., Thatcher J.W. (eds.) Complexity and Computer Computations, pp. 85–104. Plenum Press, New York (1972).Google Scholar
  102. 102.
    Kiermaier M., Laue R.: Derived and residual subspace designs. Adv. Math. Commun. 9, 105–115 (2015).Google Scholar
  103. 103.
    Kiermaier M., Pavc̆ević M.O.: Intersection numbers for subspace designs. J. Comb. Des. 23, 463–480 (2015).Google Scholar
  104. 104.
    Koetter R., Médrad M.: An algebraic approach to network coding. IEEE Trans. Netw. 11, 782–795 (2003).Google Scholar
  105. 105.
    Koetter R., Kschischang F.R.: Coding for errors and erasures in random network coding. IEEE Trans. Inf. Theory 54, 3579–3591 (2008).Google Scholar
  106. 106.
    Kohnert A., Kurz S.: Construction of large constant dimension codes with a prescribed minimum distance. In: Lecture Notes Computer Science, vol. 5393, pp. 31–42 (2008).Google Scholar
  107. 107.
    Lachaud G.: The parameters of projective Reed–Muller codes. Discret. Math. 81, 217–221 (1990).Google Scholar
  108. 108.
    Lachaud G.: Number of points of plane sections and linear codes defined on algebraic varieties. In: Arithmetic, Geometry, and Coding Theory. (Luminy, France, 1993), pp. 77–104. Walter De Gruyter, Berlin (1996).Google Scholar
  109. 109.
    Landjev I., Storme L.: Galois geometries and coding theory. In: De Beule J., Storme L. (eds.) Current Research Topics in Galois Geometry, pp. 187–214. NOVA Academic Publishers, New York (2012).Google Scholar
  110. 110.
    Li S.-Y.R., Yeung R.W., Cai N.: Linear network coding. IEEE Trans. Inf. Theory 49, 371–381 (2003).Google Scholar
  111. 111.
    Liu S., Oggier F.: On the design of orbit storage codes. In: 4th International Castle Meeting on Coding Theory and Applications, Palmela, Spain (2014).Google Scholar
  112. 112.
    Lovász L.: On two minimax theorems in graph theory. J. Comb. Theory Ser. B 21, 96–103 (1976).Google Scholar
  113. 113.
    Lunardon G.: Normal spreads. Geom. Dedicata 75, 245–261 (1999).Google Scholar
  114. 114.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).Google Scholar
  115. 115.
    Manganiello F., Gorla E., Rosenthal J.: Spread codes and spread decoding in network coding. In: International Symposium on Information Theory, pp. 881–885 (2008).Google Scholar
  116. 116.
    Menger K.: Zur allgemeinen kurventheorie. Fund Math. 10, 96–115 (1927).Google Scholar
  117. 117.
    Metsch K.: Bose–Burton type theorems for finite projective, affine and polar spaces. In: Lamb J.D., Preece D.A. (eds.) Surveys in Combinatorics 1999. London Mathematical Society Lecture Note Series, vol. 267, pp. 137–166. Cambridge University Press, Cambridge (1999).Google Scholar
  118. 118.
    Metsch K.: Blocking sets in projective spaces and polar spaces. J. Geom. 76, 216–232 (2003).Google Scholar
  119. 119.
    Metsch K.: Blocking subspaces by lines in PG(\(n, q\)). Combinatorica 24, 459–486 (2004).Google Scholar
  120. 120.
    Miyakawa M., Munemasa A., Yoshiara S.: On a class of small 2-designs over GF(\(q\)). J. Comb. Des. 3, 61–77 (1995).Google Scholar
  121. 121.
    Motwani R., Raghavan P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995).Google Scholar
  122. 122.
    Nakić A., Pavc̆ević M.O.: Tactical decompositions of designs over finite fields. Des. Codes Cryptogr. 77, 49–60 (2015).Google Scholar
  123. 123.
    Oggier F.: Some Constructions of Storage Codes from Grassmann Graphs. ETH-Zurich, Zurich (2014). doi: 10.3929/ethz-a-010094830.
  124. 124.
    Oggier F., Datta A.: Self-repairing codes for distributed storage - A projective geometric construction. In: Information Theory Workshop (ITW), pp. 30–34 (2011).Google Scholar
  125. 125.
    Olmez O., Ramamoorthy A.: Fractional repetition codes with flexible repair from combinatorial designs. arXiv:1408.5780v1 (August 2014).
  126. 126.
    Penttila T., Williams B.: Regular Packings of PG(\(3, q\)). Eur. J. Comb. 19, 713–720 (1998).Google Scholar
  127. 127.
    Pepe V., Storme L.: The use of blocking sets in Galois geometries and in related research areas. In: Narasimha, Sastry N.S. (ed.). Springer Proceedings in Mathematics. Proceedings of the Satellite Conference Buildings, Finite Geometries and Groups of the International Congress of Mathematicians 2010, Bangalore, India (29–31, August, 2010), vol. 10, pp. 305–327 (2012).Google Scholar
  128. 128.
  129. 129.
    Rashmi K.V., Shah N.B., Kumar P.V., Ramchandran K.: Explicit construction of optimal exact regenerating codes for distributed storage. In: 47th Annual Allerton Conference on Communications, Control and Computing, pp. 1243–1249 (2009).Google Scholar
  130. 130.
    Raviv N., Etzion T.: Distributed storage systems based on intersecting subspace codes. In: International Symposium on Information Theory, pp. 1462–1466 (2015). arXiv:1406.6170 (June 2014).
  131. 131.
    Riis S., Ahlswede R.: Problems in Network Coding and Error Correcting Codes. Appended by a draft version of Riis S.: Utilising public information in network coding. General Theory of Information Transfer and Combinatorics. In: Lecture Notes in Computer Science, vol. 4123, pp. 861–897 (lecture 3) (2006).Google Scholar
  132. 132.
    Rodier F., Sboui A.: Les arrangements minimaux et maximaux d’hyperplans dans \(\mathbb{P}^{n}(\mathbb{F}_{q})\). C. R. Acad. Sci. Paris Ser. I 344, 287–290 (2007).Google Scholar
  133. 133.
    Rosenthal J., Silberstein N., Trautmann A.-L.: On the geometry of balls in the Grassmannian and list decoding of lifted Gabidulin codes. Des. Codes Cryptogr. 73, 393–416 (2014).Google Scholar
  134. 134.
    Roth R.M.: Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Inf. Theory 37, 328–336 (1991).Google Scholar
  135. 135.
    Sarmiento J.: Resolutions of PG(5,2) with point-cyclic automorphism group. J. Comb. Des. 8, 2–14 (2000).Google Scholar
  136. 136.
    Sarmiento J.: On point-cyclic resolutions of the \(2-(63,7,15)\) design associated with PG(5,2). Gr. Comb. 18, 621–632 (2002).Google Scholar
  137. 137.
    Sboui A.: Special numbers of rational points on hypersurfaces in the \(n\)-dimensional projective space over a finite field. Discret. Math. 309, 5048–5059 (2009).Google Scholar
  138. 138.
    Schönheim J.: On coverings. Pac. J. Math. 14, 1405–1411 (1964).Google Scholar
  139. 139.
    Schwartz M., Etzion T.: Codes and anticodes in the Grassman graph. J. Comb. Theory, Ser. A 97, 27–42 (2002).Google Scholar
  140. 140.
    Segre B.: Ovals in a finite projective plane. Can. J. Math. 7, 414–416 (1955).Google Scholar
  141. 141.
    Serre J.-P.: Lettre à M. Tsfasman du 24 Juillet 1989. Journées Arithmétiques de Luminy 17-21 Juillet 1989. Astérisque 198-199-200, 11 (1991), 351–353 (1992).Google Scholar
  142. 142.
    Silberstein N., Etzion T.: Fractional repetition codes for repair in distributed storage systems. arXiv:1401.4734v3 (January 2014)
  143. 143.
    Silva D., Kschischang F.R.: On metrics for error-correction in network coding. IEEE Trans. Inf. Theory 55, 5479–5490 (2009).Google Scholar
  144. 144.
    Silva D., Kschischang F.R., Koetter R.: A rank-metric approach to error control in random network coding. IEEE Trans. Inf. Theory 54, 3951–3967 (2008).Google Scholar
  145. 145.
    Skachek V.: Recursive code construction for random networks. IEEE Trans. Inf. Theory 56, 1378–1382 (2010).Google Scholar
  146. 146.
    Solomon G., Stiffler J.J.: Algebraically punctured cyclic codes. Inf. Control 8, 170–179 (1965).Google Scholar
  147. 147.
    Sørensen A.B.: Projective Reed–Muller codes. IEEE Trans. Inf. Theory 37, 1567–1576 (1991).Google Scholar
  148. 148.
    Suzuki H.: 2-Designs over GF(\(2^m\)). Gr. Comb. 6, 293–296 (1990).Google Scholar
  149. 149.
    Suzuki H.: On the inequalities of \(t\)-designs over a finite field. Eur. J. Comb. 11, 601–607 (1990).Google Scholar
  150. 150.
    Suzuki H.: 2-Designs over GF(\(q\)). Gr. Comb. 8, 381–389 (1992).Google Scholar
  151. 151.
    Sziklai P.: On small blocking sets and their linearity. J. Comb. Theory Ser. A 115, 1167–1182 (2008).Google Scholar
  152. 152.
    Szőnyi T.: Blocking sets in Desarguesian affine and projective planes. Finite Fields Appl. 3, 187–202 (1997).Google Scholar
  153. 153.
    Szőnyi T., Weiner Zs.: Small blocking sets in higher dimensions. J. Comb. Theory Ser. A 95, 88–101 (2001).Google Scholar
  154. 154.
    Thomas S.: Designs over finite fields. Geom. Dedicata 21, 237–242 (1987).Google Scholar
  155. 155.
    Thomas S.: Designs and partial geometries over finite fields. Geom. Dedicata 63, 247–253 (1996).Google Scholar
  156. 156.
    Tits J.: Sur les analogues algébriques des groupes semi-simples complexes. Colloque d’Algèbre Supérieure, tenu á Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain. Librairie Gauthier-Villars, Paris, pp. 261–289 (1957).Google Scholar
  157. 157.
    Topalova S., Zhelezova S.: 2-spreads and transitive and orthogonal 2-parallelisms of PG(5,2). Gr. Comb. 26, 727–735 (2010).Google Scholar
  158. 158.
    Trautmann A.-L.: Isometry and automorphisms of constant dimension codes. Adv. Math. Commun. 7, 147–160 (2013).Google Scholar
  159. 159.
    Trautmann A.-L., Rosenthal J.: A complete characterization of irreducible cyclic orbit codes and their Plücker embedding. Des. Codes Cryptogr. 66, 275–289 (2013).Google Scholar
  160. 160.
    Trautmann A.-L., Manganiello F., Braun M., Rosenthal J.: Cyclic orbit codes. IEEE Trans. Inf. Theory 59, 7386–7404 (2013).Google Scholar
  161. 161.
    Wang J.: Quotient sets and subset-subspace analogy. Adv. Appl. Math. 23, 333–339 (1999).Google Scholar
  162. 162.
    Wang H., Xing C., Safavi-Naini R.M.: Linear authentication codes: bounds and constructions. IEEE Trans. Inf. Theory 49, 866–872 (2003).Google Scholar
  163. 163.
    Wachter-Zeh A., Etzion T.: Optimal Ferrers Diagram Rank-Metric Codes. arXiv:1405.1885 (May 2014).
  164. 164.
    Xia S.-T., Fu F.-W.: Johnson type bounds on constant dimension codes. Des. Codes Cryptogr. 50, 163–172 (2009).Google Scholar
  165. 165.
    Zaicev G.V., Zinoviev V.A., Semakov N.V.: Interrelations of Preparata and Hamming codes and extension of Hamming codes to new double error-correcting codes. In: Proceedings of the 2nd International Symposium on Information Theory, Budapest, pp. 257–263 (1971).Google Scholar
  166. 166.
    Zhang Z.: Theory and applications of network error correction coding. Proc. IEEE 99, 406–420 (2011).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Computer Science DepartmentTechnion IITHaifaIsrael
  2. 2.Department of MathematicsGhent UniversityGhentBelgium

Personalised recommendations