[go: up one dir, main page]
Skip to main content
SpringerLink
Log in

Search for Combinatorial Objects Using Lattice Algorithms – Revisited

  • Conference paper
  • First Online:
Combinatorial Algorithms (IWOCA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12757))

Included in the following conference series:

Abstract

In 1986, Kreher and Radziszowski were the first who used the famous LLL algorithm to construct combinatorial designs. Subsequently, lattice algorithms have been applied to construct a large variety of objects in design theory, coding theory and finite geometry. Unfortunately, the use of lattice algorithms in combinatorial search is still not well established. Here, we provide a list of problems which could be tackled with this approach and give an overview on exhaustive search using lattice basis reduction. Finally, we describe a different enumeration strategy which might improve the power of this method even further.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Updated versions available at https://www-cs-faculty.stanford.edu/~knuth/programs.html.

References

  1. Aardal, K., Hurkens, C., Lenstra, A.K.: Solving a linear diophantine equation with lower and upper bounds on the variables. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds.) IPCO 1998. LNCS, vol. 1412, pp. 229–242. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-69346-7_18

    Chapter  MATH  Google Scholar 

  2. Betten, A., Kerber, A., Laue, R., Wassermann, A.: Simple 8-designs with small parameters. Des. Codes Crypt. 15, 5–27 (1998)

    Article  MathSciNet  Google Scholar 

  3. Betten, A., Kerber, A., Kohnert, A., Laue, R., Wassermann, A.: The discovery of simple 7-designs with automorphism group \(P{\Gamma }L\)(2, 32). In: Cohen, G., Giusti, M., Mora, T. (eds.) AAECC 1995. LNCS, vol. 948, pp. 131–145. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60114-7_10

    Chapter  Google Scholar 

  4. Betten, A., Klin, M., Laue, R., Wassermann, A.: Graphical \(t\)-designs. Discrete Math. 197(198), 111–121 (1999)

    Article  MathSciNet  Google Scholar 

  5. Betten, A., Laue, R., Wassermann, A.: New \(t\)-designs and large sets of \(t\)-designs. Discrete Math. 197(198), 83–109 (1999)

    Article  MathSciNet  Google Scholar 

  6. Bouyukliev, I., Bouyuklieva, S., Kurz, S.: Computer classification of linear codes. CoRR abs/2002.07826 (2020). https://arxiv.org/abs/2002.07826

  7. Braun, M., Kohnert, A., Wassermann, A.: Construction of \((n, r)\)-arcs in PG(\(2, q\)). Innovations Incidence Geom. 1, 133–141 (2005)

    Article  MathSciNet  Google Scholar 

  8. Braun, M., Kerber, A., Laue, R.: Systematic construction of \(q\)-analogs of \(t\)-\((v, k,\lambda )\)-designs. Des. Codes Crypt. 34(1), 55–70 (2005). https://doi.org/10.1007/s10623-003-4194-z

    Article  MathSciNet  MATH  Google Scholar 

  9. Braun, M., Kiermaier, M., Kohnert, A., Laue, R.: Large sets of subspace designs. J. Comb. Theory Ser. A 147, 155–185 (2017). https://doi.org/10.1016/j.jcta.2016.11.004

    Article  MathSciNet  MATH  Google Scholar 

  10. Braun, M., Kiermaier, M., Wassermann, A.: Computational methods in subspace designs. In: Greferath, M., Pavčević, M.O., Silberstein, N., Vázquez-Castro, M.Á. (eds.) Network Coding and Subspace Designs. SCT, pp. 213–244. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-70293-3_9

    Chapter  Google Scholar 

  11. Braun, M., Kohnert, A., Östergård, P.R.J., Wassermann, A.: Large sets of \(t\)-designs over finite fields. J. Comb. Theory A 124, 195–202 (2014)

    Article  MathSciNet  Google Scholar 

  12. Braun, M., Kohnert, A., Wassermann, A.: Optimal linear codes from matrix groups. IEEE Trans. Inform. Theory 51(12), 4247–4251 (2005). https://doi.org/10.1109/TIT.2005.859291

    Article  MathSciNet  MATH  Google Scholar 

  13. Buratti, M., Kiermaier, M., Kurz, S., Nakić, A., Wassermann, A.: \(q\)-analogs of group divisible designs. In: Pseudorandomness and Finite Fields, Radon Series on Computational and Applied Mathematics, vol. 23. DeGruyter (2019)

    Google Scholar 

  14. Buratti, M., Wassermann, A.: On decomposability of cyclic triple systems. Australas. J. Comb. 71(2), 184–195 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Berlin (1993). https://doi.org/10.1007/978-3-662-02945-9

    Book  MATH  Google Scholar 

  16. Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs, Second Edition (Discrete Mathematics and Its Applications). Chapman & Hall/CRC, Boca Raton (2006)

    Google Scholar 

  17. Coster, M., Joux, A., LaMacchia, B., Odlyzko, A., Schnorr, C., Stern, J.: Improved low-density subset sum algorithms. Comput. Complex. 2, 111–128 (1992)

    Article  MathSciNet  Google Scholar 

  18. Coster, M.J., LaMacchia, B.A., Odlyzko, A.M., Schnorr, C.P.: An improved low-density subset sum algorithm. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 54–67. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-46416-6_4

    Chapter  Google Scholar 

  19. Coveyou, R., MacPherson, R.: Fourier analysis of uniform random number generators. J. ACM 14, 100–119 (1967)

    Article  MathSciNet  Google Scholar 

  20. Dieter, U.: How to calculate shortest vectors in a lattice. Math. Comput. 29(131), 827–833 (1975)

    Article  MathSciNet  Google Scholar 

  21. Fincke, U., Pohst, M.: Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comput. 44, 463–471 (1985)

    Article  MathSciNet  Google Scholar 

  22. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  23. Gibbons, P.B., Östergård, P.R.J.: Computational methods in design theory. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, chap. VII.6, 2 edn, pp. 755–783. Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  24. Gurobi Optimization: Gurobi optimizer reference manual (2016). http://www.gurobi.com

  25. Harvey, W.D., Ginsberg, M.L.: Limited discrepancy search. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence, IJCAI 1995, vol. 1. pp. 607–613. Morgan Kaufmann Publishers Inc., San Francisco (1995)

    Google Scholar 

  26. Hermite, C.: Extraits de lettres de M.Ch. Hermite à M. Jacobi sur différents objets de la théorie des nombres. J. reine angew. Math. 40, 279–290 (1850)

    Google Scholar 

  27. IBM: ILOG CPLEX Optimizer (2010)

    Google Scholar 

  28. Kaib, M., Ritter, H.: Block reduction for arbitrary norms. Preprint, Universität Frankfurt (1995)

    Google Scholar 

  29. Kannan, R.: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12, 415–440 (1987)

    Article  MathSciNet  Google Scholar 

  30. Karoui, W., Huguet, M.-J., Lopez, P., Naanaa, W.: YIELDS: a yet improved limited discrepancy search for CSPs. In: Van Hentenryck, P., Wolsey, L. (eds.) CPAIOR 2007. LNCS, vol. 4510, pp. 99–111. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72397-4_8

    Chapter  MATH  Google Scholar 

  31. Kaski, P., Östergård, P.R.: Classification Algorithms for Codes and Designs. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-28991-7

  32. Kiermaier, M., Kurz, S., Solé, P., Stoll, M., Wassermann, A.: On strongly walk regular graphs, triple sum sets and their codes. ArXiv e-prints, abs/1502.02711 (2020)

    Google Scholar 

  33. Kiermaier, M., Laue, R., Wassermann, A.: A new series of large sets of subspace designs over the binary field. Des. Codes Crypt. 86(2), 251–268 (2018). https://doi.org/10.1007/s10623-017-0349-1

    Article  MathSciNet  MATH  Google Scholar 

  34. Kiermaier, M., Wassermann, A., Zwanzger, J.: New upper bounds on binary linear codes and a \({\mathbb{Z}}_{4}\)-code with a better-than-linear Gray image. IEEE Trans. Inf. Theory 62(12), 6768–6771 (2016). https://doi.org/10.1109/TIT.2016.2612654

    Article  MathSciNet  MATH  Google Scholar 

  35. Knuth, D.E.: Dancing links. In: Davies, J., Roscoe, B., Woodcock, J. (eds.) Millennial Perspectives in Computer Science: Proceedings of the 1999 Oxford-Microsoft Symposium in Honour of Sir Tony Hoare. Palgrave (2000)

    Google Scholar 

  36. Knuth, D.: The Art of Computer Programming, Vol. 2: Seminumerical Algorithms. Addison-Wesley, Reading (1969)

    Google Scholar 

  37. Kohnert, A.: Constructing two-weight codes with prescribed groups of automorphisms. Discret. Appl. Math. 155(11), 1451–1457 (2007). https://doi.org/10.1016/j.dam.2007.03.006

    Article  MathSciNet  MATH  Google Scholar 

  38. Korkine, A., Zolotareff, G.: Sur les formes quadratiques. Math. Ann. 6, 366–389 (1873)

    Article  MathSciNet  Google Scholar 

  39. Kramer, E.S., Mesner, D.M.: \(t\)-designs on hypergraphs. Discret. Math. 15(3), 263–296 (1976). https://doi.org/10.1016/0012-365X(76)90030-3

    Article  MathSciNet  MATH  Google Scholar 

  40. Kreher, D.L., Radziszowski, S.P.: The existence of simple \(6\)-\((14,7,4)\) designs. J. Comb. Theory Ser. A 43, 237–243 (1986)

    Google Scholar 

  41. Kreher, D.L., Radziszowski, S.P.: Finding simple \(t\)-designs by using basis reduction. Congr. Numer. 55, 235–244 (1986)

    MathSciNet  MATH  Google Scholar 

  42. Krčadinac, V.: Some new designs with prescribed automorphism groups. J. Comb. Des. 26(4), 193–200 (2018). https://doi.org/10.1002/jcd.21587

    Article  MathSciNet  MATH  Google Scholar 

  43. Krčadinac, V., Pavčević, M.O.: New small 4-designs with nonabelian automorphism groups. In: Blömer, J., Kotsireas, I.S., Kutsia, T., Simos, D.E. (eds.) MACIS 2017. LNCS, vol. 10693, pp. 289–294. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-72453-9_23

    Chapter  Google Scholar 

  44. Lagarias, J., Odlyzko, A.: Solving low-density subset sum problems. J. Assoc. Comp. Mach. 32, 229–246 (1985). Appeared already in Proceedings of 24th IEEE Symposium on Foundations of Computer Science, pp. 1–10 (1983)

    Google Scholar 

  45. Laue, R.: Constructing objects up to isomorphism, simple 9-designs with small parameters. In: Betten, A., Kohnert, A., Laue, R., Wassermann, A. (eds.) Algebraic Combinatorics and Applications, pp. 232–260. Springer, Heidelberg (2001). https://doi.org/10.1007/978-3-642-59448-9_16

    Chapter  Google Scholar 

  46. Laue, R., Magliveras, S., Wassermann, A.: New large sets of t-designs. J. Comb. Des. 9, 40–59 (2001)

    Article  MathSciNet  Google Scholar 

  47. Laue, R., Omidi, G.R., Tayfeh-Rezaie, B., Wassermann, A.: New large sets of \(t\)-designs with prescribed groups of automorphisms. J. Combin. Des. 15(3), 210–220 (2007). https://doi.org/10.1002/jcd.20128

    Article  MathSciNet  MATH  Google Scholar 

  48. Lenstra, A., Lenstra Jr., H., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)

    Article  MathSciNet  Google Scholar 

  49. Mathon, R.: Computational methods in design theory. In: Keedwell, A.D. (ed.) Surveys in Combinatorics, Proc. 13th Br. Comb. Conf., Guildford/UK 1991, vol. 166, pp. 101–117. London Mathematical Society Lecture Note (1991)

    Google Scholar 

  50. Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems. Kluwer Academic Publishers (2002)

    Google Scholar 

  51. Minkowski, H.: Geometrie der Zahlen. Teubner, Leipzig (1896)

    Google Scholar 

  52. Nguyen, P.Q., Vallée, B.: The LLL Algorithm: Survey and Applications, 1st edn. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02295-1

    Book  MATH  Google Scholar 

  53. Niskanen, S., Östergård, P.R.J.: Cliquer user’s guide, version 1.0. Technical report T48, Helsinki University of Technology (2003)

    Google Scholar 

  54. Östergård, P.R.J., Quistorff, J., Wassermann, A.: New results on codes with covering radius 1 and minimum distance 2. Des. Codes Crypt. 35, 241–250 (2005)

    Article  MathSciNet  Google Scholar 

  55. Ritter, H.: Aufzählung von kurzen Gittervektoren in allgemeiner Norm. Ph.D. thesis, Universität Frankfurt (1997)

    Google Scholar 

  56. Schnorr, C.: A hierachy of polynomial time lattice basis reduction algorithms. Theoret. Comput. Sci. 53, 201–224 (1987)

    Article  MathSciNet  Google Scholar 

  57. Schnorr, C.P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. In: Budach, L. (ed.) FCT 1991. LNCS, vol. 529, pp. 68–85. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54458-5_51

    Chapter  Google Scholar 

  58. Schnorr, C.P., Hörner, H.H.: Attacking the Chor-Rivest cryptosystem by improved lattice reduction. In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 1–12. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-49264-X_1

    Chapter  Google Scholar 

  59. van Beek, P.: Backtracking search algorithms. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, Foundations of Artificial Intelligence, vol. 2, pp. 85–134. Elsevier (2006). https://doi.org/10.1016/S1574-6526(06)80008-8

  60. Wassermann, A.: Finding simple \(t\)-designs with enumeration techniques. J. Comb. Des. 6(2), 79–90 (1998)

    Article  MathSciNet  Google Scholar 

  61. Wassermann, A.: Attacking the market split problem with lattice point enumeration. J. Comb. Optim. 6, 5–16 (2002)

    Article  MathSciNet  Google Scholar 

  62. Wassermann, A.: Computing the minimum distance of linear codes. In: Eighth International Workshop Algebraic and Combinatorial Coding Theory (ACCT VIII), Tsarskoe Selo, Russia, pp. 254–257 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bayreuth, 95440, Bayreuth, Germany

    Alfred Wassermann

Authors
  1. Alfred Wassermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alfred Wassermann .

Editor information

Editors and Affiliations

  1. University of Ottawa, Ottawa, ON, Canada

    Paola Flocchini

  2. University of Ottawa, Ottawa, ON, Canada

    Lucia Moura

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wassermann, A. (2021). Search for Combinatorial Objects Using Lattice Algorithms – Revisited. In: Flocchini, P., Moura, L. (eds) Combinatorial Algorithms. IWOCA 2021. Lecture Notes in Computer Science(), vol 12757. Springer, Cham. https://doi.org/10.1007/978-3-030-79987-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-79987-8_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-79986-1

  • Online ISBN: 978-3-030-79987-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation