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Bounds on the sizes of constant weight covering codes

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Abstract

Motivated by applications in universal data compression algorithms we study the problem of bounds on the sizes of constant weight covering codes. We are concerned with the minimal sizes of codes of lengthn and constant weightu such that every word of lengthn and weightv is within Hamming distanced from a codeword. In addition to a brief summary of part of the relevant literature, we also give new results on upper and lower bounds to these sizes. We pay particular attention to the asymptotic covering density of these codes. We include tables of the bounds on the sizes of these codes both for small values ofn and for the asymptotic behavior. A comparison with techniques for attaining bounds for packing codes is also given. Some new combinatorial questions are also arising from the techniques.

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References

  1. E. D. Boyer, D. L. Kreher, S. P. Radziszowski, J. Zou, and A. Sidorenko, On (n, 5, 3)-Turán systems, preprint.

  2. A. E. Brouwer, J. B. Shearer, N. J. A. Sloane, and W. D. Smith, A new table of constant weight codes.IEEE Trans. on Information Theory, Vol. IT-36 (1990) pp. 1334–1380.

    Google Scholar 

  3. A. E. Brouwer and M. Voorhoeve, Turán theory and lotto problem.Math. Centre Tracts, Vol. 106 (1979) pp. 99–105.

    Google Scholar 

  4. D. de Caen, On Turán's Hypergraph problem, Ph.D. thesis, University of Toronto (1982).

  5. D. de Caen, A note on the probabilistic approach to Turán's problem.J. of Combinatorial Theory, Ser. B, Vol. 34 (1983), pp. 340–349.

    Google Scholar 

  6. D. de Caen, Extension of a theorem of Moon and Moser on complete subgraphs,Ars Combinatoria, Vol. 16 (1983), pp. 5–10.

    Google Scholar 

  7. D. de Caen, D. L. Kreher, and J. Wiseman. On constructive upper bounds for the Turán numbers.Congressus Numerantium, Vol. 65 (1988), pp. 277–280.

    Google Scholar 

  8. D. de Caen, D. L. Kreher, S. P. Radziszowski, and W. H. Mills, On the covering oft-sets with (t+1)-sets:C(9, 5, 4) andC(10, 6, 5).Discrete Mathematics, Vol. 92 (1991) pp. 65–77.

    Google Scholar 

  9. G. D. Cohen, M. G. Karpovsky, H. F. Mattson Jr., and J. R. Schatz, Covering radius—survey and recent results,IEEE Trans. on Information Theory, Vol. IT-31 (1985) pp. 328–343.

    Google Scholar 

  10. T. Etzion, Large sets of coverings,J of Combinatorial Design, vol. 2 (1994), 359–373.

    Google Scholar 

  11. M. K. Fort Jr. and G. A. Hedlund, Minimal coverings of pairs by triples,Pacific J. Math., Vol. 8 (1958) pp. 709–717.

    Google Scholar 

  12. R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes,IEEE Trans. on Information Theory, Vol. IT-26 (1980) pp. 37–43.

    Google Scholar 

  13. R. L. Gray and L. D. Davisson, Source coding theorems without the ergodic assumption.IEEE Trans. on Information Theory, Vol. IT-20, (1974) pp. 502–516.

    Google Scholar 

  14. H. Hanani, D. Ornstein, and V. T. Sos, On the lottery problem,Magyer Tud. Akad. Mat. Kutato Int. Kozl., Vol. 9 (1964) pp. 155–158.

    Google Scholar 

  15. H. Hämäläinen and S. Rankinen, Upper bounds for football pool problems and mixed covering codes.J. of Combinatorial Theory, Ser. A, Vol. 56 (1991) pp. 84–95.

    Google Scholar 

  16. A. Hartman, W. H. Mills, and R. C. Mullin, Covering triples by quadruples,J. of Combinatorial Theory, Ser. A, Vol. 41 (1986) pp. 117–138.

    Google Scholar 

  17. I. S. Honkala, Modified bounds for covering codes.IEEE Trans. on Information Theory, Vol. IT-37 (1991) pp. 351–365.

    Google Scholar 

  18. C. C. Lindner and A. Rosa, Constructions of large sets of almost disjoint Steiner triple systems,J. Canad. Math., Vol. 27 (1974) pp. 256–260.

    Google Scholar 

  19. J. X. Lu, On large sets of disjoint Steiner triple systems I–III,J. of Combinatorial Theory, Ser. A, Vol. 34 (1983) pp. 140–146, 147–155, 156–183.

    Google Scholar 

  20. J. X. Lu, On large sets of disjoint Steiner triple systems IV–VI,J. of Combinatorial Theory, Ser. A, Vol. 37 (1984) pp. 136–163, 164–188, 189–192.

    Google Scholar 

  21. W. H. Mills, Covering problems,Proc. 4th S-E Conf. Combinatorics, Graph Theory, and Computing, (1973) pp. 23–52.

  22. W. H. Mills, Constructions of covering designs,Congressus Numerantium, Vol. 73 (1990) pp. 29–36.

    Google Scholar 

  23. V. Rödl, On a packing and covering problem,Europ. J. Combin., Vol. 6 (1985) pp. 69–78.

    Google Scholar 

  24. J. Schönheim, On Coverings,Pacific. J. Math., Vol. 14 (1964) pp. 1405–1411.

    Google Scholar 

  25. T. H. Spencer, On the size of independent sets in hypergraphs, Marshall Hall conference on Codes, Designs, and Groups, University of Vermont, September 13–18 (1990).

  26. R. G. Stanton and J. A. Bate, A computer search for B-coverings, Combinatorial Math. VII, Lecture Notes in Math., Springer-Verlag, New York, 829 (1981) pp. 37–50.

    Google Scholar 

  27. R. G. Stanton, J. G. Kalbfleisch, and R. C. Mullin, Covering and packing designs, Proceedings of the Second Chapel Hill Conference on Combinatorial Mathematics and its Applications, University of North Carolina, Chapel Hill, NC (1970) pp. 428–450.

  28. L. Teirlinck, A completion of Lu's determination of the spectrum for large sets of disjoint Steiner systems,J. of Combinatoridl Theory, Ser. A, Vol. 57 (1991) pp. 302–305.

    Google Scholar 

  29. P. Turán, Eine extremalaufgabe aus der Graphentheorie,Math. Fiz. Lapok, Vol. 48 (1941) pp. 436–452.

    Google Scholar 

  30. P. Turán, On the theory of graphs,Colloq. Math., Vol. 3 (1954) pp. 146–163.

    Google Scholar 

  31. P. Turán, Research problems,Kozl. MTA Mat. Kutato Int., Vol. 6 (1961) pp. 417–423.

    Google Scholar 

  32. C. L. van Pul and T. Etzion, New lower bounds for constant weight codes,IEEE Trans. on Inform. Theory, Vol. IT-35 (1989) pp. 1324–1329.

    Google Scholar 

  33. V. Wei and Z. Zhang, manuscript in preparation.

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Authors and Affiliations

  1. Computer Science Department, Technion, 32000, Haifa, Israel

    Tuvi Etzion

  2. Bellcore, 445 South Street, 07960, Morristown, NJ, USA

    Victor Wei

  3. Department of Electrical Engineering, University of Southern California, 90089, Los Angeles, CA, USA

    Zhen Zhang

Authors
  1. Tuvi Etzion
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  2. Victor Wei
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  3. Zhen Zhang
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Additional information

Part of this work was done while the first and third authors were visiting Bellcore. The third author was supported in part by National Science Foundation under grant NCR-8905052. Part of this work was presented in the Coding and Quantization Workshop at Rutgers University, NJ, October 1992.

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Etzion, T., Wei, V. & Zhang, Z. Bounds on the sizes of constant weight covering codes. Des Codes Crypt 5, 217–239 (1995). https://doi.org/10.1007/BF01388385

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  • DOI: https://doi.org/10.1007/BF01388385

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