Skip to main content
SpringerLink
Log in

A construction for strength-3 covering arrays from linear feedback shift register sequences

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We present a new construction for covering arrays inspired by ideas from Munemasa (Finite Fields Appl 4:252–260, 1998) using linear feedback shift registers (LFSRs). For a primitive polynomial \(f\) of degree \(m\) over \(\mathbb F _q\), by taking all unique subintervals of length \(\frac{q^m-1}{q-1}\) from the LFSR generated by \(f\), we derive a general construction for optimal variable strength orthogonal arrays over an infinite family of abstract simplicial complexes. For \(m=3\), by adding the subintervals of the reversal of the LFSR to the variable strength orthogonal array, we derive a strength-3 covering array over \(q^2+q+1\) factors, each with \(q\) levels that has size only \(2q^3-1\), i.e. a \(\text {CA}(2q^3-1; 3, q^2+q+1, q)\) whenever \(q\) is a prime power. When \(q\) is not a prime power, we obtain results by using fusion operations on the constructed array for higher prime powers and obtain improved bounds. Colbourn maintains a repository of the best known bounds for covering array sizes for all \(2 \le q \le 25\). Our construction, with fusing when applicable, currently holds records of the best known upper bounds in this repository for all \(q\) except \(q = 2,3,6\). By using these covering arrays as ingredients in recursive constructions, we build covering arrays over larger numbers of factors, again providing significant improvements on the previous best upper bounds.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Ahmed B.S., Zamli K.Z.: A variable strength interaction test suites generation strategy using particle swarm optimization. J. Syst. Softw. 84, 2171–2185 (2011).

    Google Scholar 

  2. Ahmed B.S., Zamli K.Z., Lim C.P.: Application of particle swarm optimization to uniform and variable strength covering array construction. Appl. Softw. Comput. 12, 1330–1347 (2012).

    Google Scholar 

  3. Bryce R.C., Colbourn C.J.: A density-based greedy algorithm for higher strength covering arrays. Softw. Test. Verification Reliab. 19(1), 37–53 (2009).

    Google Scholar 

  4. Burr K., Young W.: Combinatorial test techniques: table-based automation, test generation and code coverage. In: International Conference on Software Testing Analysis and Review, San Diego, CA (1998).

  5. Burroughs K., Jain A., Erickson R.L.: Improved quality of protocol testing through techniques of experimental design. In: Supercomm/IC, IEEE International Conference on Communications, pp. 745–752 (1994).

  6. Chateauneuf M., Kreher D.: On the state of strength-three covering arrays. J. Comb. Des. 10(4), 217–238 (2002).

    Google Scholar 

  7. Chen X., Gu Q., Li A., Chen D.: Variable strength interaction testing with an ant colony system approach. In: I.C. Society (ed.) 2009 16th Asia-Pacific Sofware Engineering Conference, pp. 160–167 (2009).

  8. Cohen D.M., Dalal S.R., Parelius J., Patton G.C.: The combinatorial design approach to automatic test generation. IEEE Softw. 13(5), 83–88 (1996).

    Google Scholar 

  9. Cohen M.B., Colbourn C.J., Collofello J.S., Gibbons P.B., Mugridge W.B.: Variable strength interaction testing of components. In: Proceedings of the 27th International Computer Software and Applications Conference (COMPSAC 2003), Dallas, TX, pp. 413–418 (2003).

  10. Cohen M.B., Colbourn C.J., Ling A.C.H.: Constructing strength three covering arrays with augmented annealing. Discret. Math. 308, 2709–2722 (2008).

    Google Scholar 

  11. Colbourn C.J.: Covering array tables. http://www.public.asu.edu/ccolbou/src/tabby/catable.html. Accessed 26 Dec 2012.

  12. Colbourn C.J.: Combinatorial aspects of covering arrays. Le Matematiche (Catania) 58, 121–167 (2004).

    Google Scholar 

  13. Colbourn C.J.: Distributing hash families and covering arrays. J. Comb. Inf. Syst. Sci. 34, 113–126 (2009).

    Google Scholar 

  14. Colbourn C.J.: Covering arrays and hash families. In: Information Security, Coding Theory and Related Combinatorics, pp. 99–135. IOS Press, Amsterdam (2011).

  15. Colbourn C.J.: Resolvable covering arrays. J. Stat. Theory Pract. (to appear)

  16. Colbourn C.J., Kéri G., Soriano P.P.R., Schlage-Puchta J.C.: Covering and radius-covering arrays: constructions and classification. Discret. Appl. Math. 158, 1158–1180 (2010).

    Google Scholar 

  17. Colbourn C.J., Martirosyan S.S., Trung T.V., Walker II R.A.: Roux-type constructions for covering arrays of strengths three and four. Des. Codes Cryptogr. 41(1), 33–57 (2006).

    Google Scholar 

  18. Dalal S.R., Mallows C.L.: Factor-covering designs for testing software. Technometrics 40(3), 234–243 (1998).

    Google Scholar 

  19. Dunietz I.S., Ehrlich W.K., Iannino A., Mallows C.L., Szablak B.D.: Applying designs of experiments to software testing. In: Proceedings of the International Conference on Software Engineering (ICSE 97), New York, NY, pp. 205–215 (1997).

  20. Golomb S.W., Gong G.: Signal Design for Good Correlation for Wireless Communication, Cryptography, and Radar. Cambridge University Press, Cambridge (2005).

  21. Hedayat A.S., Sloane N.J.A., Stufken J.: Orthogonal Arrays. Springer Series in Statistics. Springer, New York (1999).

  22. Kuhn D.R., Wallace D.R., Gallo Jr A.M.: Software fault interactions and implications for software testing. IEEE Trans. Softw. Eng. 30(6), 418–421 (2004).

    Google Scholar 

  23. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997)

  24. Maity Y., Maity S.: Mixed covering arrays on hypergraphs. Commun. Comput. Inf. Sci. 305, 327–338 (2012).

    Google Scholar 

  25. Meagher K.: Covering arrays on graphs: qualitative independence graphs and extremal set-partition theory. Ph.D. thesis, University of Ottawa (2005).

  26. Meagher K., Stevens B.: Covering arrays on graphs. J. Comb. Theory B 95(1), 134–151 (2005).

    Google Scholar 

  27. Meagher K., Moura L., Zekaoui L.: Mixed covering arrays on graphs. J. Comb. Des. 15, 393–404 (2007).

    Google Scholar 

  28. Munemasa A.: Orthogonal arrays, primitive trinomials, and shift-register sequences. Finite Fields Appl. 4, 252–260 (1998).

    Google Scholar 

  29. Pott A.: Finite Geometry and Character Theory. Lecture Notes in Mathematics, vol. 1601. Springer, Berlin (1995).

  30. Raaphorst S.: Variable strength covering arrays. Ph.D. thesis, University of Ottawa (2013).

  31. Raaphorst S., Moura L., Stevens B.: A density-based greedy algorithm for variable strength covering arrays. Australas. J. Comb. (to appear).

  32. Wang Z., Xu B., Nie C.: Greedy heuristic algorithms to generate variable strength combinatorial test stuie. In: Proceedings of Eighth International Conference on Quality Software, QSIC ’08, pp. 155–160. IEEE Computer Society, Washington, DC (2008).

  33. Williams A.W., Probert R.L.: A measure for component interaction test coverage. In: Proceedings of the ACS/IEEE International Conference on Computer Systems and Applications, Beirut, pp. 301–311 (2001).

  34. Zierler N.: Linear recurring sequences. J. Soc. Ind. Appl. Math. 7(1), 31–48 (1959).

    Google Scholar 

Download references

Acknowledgments

The second and third authors were supported by NSERC Discovery grants. We would like to thank Charles Colbourn for his help in producing Table 3, and for providing several references for the constructions used to build it.

Author information

Authors and Affiliations

  1. School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Avenue, Ottawa, ON,  K1K 6N5, Canada

    Sebastian Raaphorst & Lucia Moura

  2. School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON,  K1S 5B6, Canada

    Brett Stevens

Authors
  1. Sebastian Raaphorst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lucia Moura
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Brett Stevens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastian Raaphorst.

Additional information

Communicated by C. J. Colbourn.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Raaphorst, S., Moura, L. & Stevens, B. A construction for strength-3 covering arrays from linear feedback shift register sequences. Des. Codes Cryptogr. 73, 949–968 (2014). https://doi.org/10.1007/s10623-013-9835-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-013-9835-2

Keywords

Mathematical Subject Classification

Search

Navigation