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Hanani triple packings and optimal \(q\)-ary codes of constant weight three

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Abstract

The exact sizes of optimal \(q\)-ary codes of length \(n\), constant weight \(w\) and distance \(d=2w-1\) have only been determined for \(q\in \{2,3\}\), and for \(w|(q-1)n\) and \(n\) sufficiently large. We completely determine the exact size of optimal \(q\)-ary codes of constant weight three and minimum distance five for all \(q\) by establishing a connection with Hanani triple packings, and settling their existence.

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References

  1. Agrell E., Vardy A., Zeger K.: Upper bounds for constant-weight codes. IEEE Trans. Inf. Theory 46(7), 2373–2395 (2000).

    Google Scholar 

  2. Blake-Wilson S., Phelps K.T.: Constant weight codes and group divisible designs. Des. Codes Cryptogr. 16(1), 11–27 (1999).

    Google Scholar 

  3. Bogdanova G.: New bounds for the maximum size of ternary constant weight codes. Serdica Math. J. 26(1), 5–12 (2000).

    Google Scholar 

  4. Brouwer A.E., Shearer J.B., Sloane N.J.A., Smith W.D.: A new table of constant weight codes. IEEE Trans. Inf. Theory 36(6), 1334–1380 (1990).

    Google Scholar 

  5. Cao H., Ji L., Zhu L.: Constructions for generalized Steiner systems GS(3,4, v,2). Des. Codes Cryptogr. 45(2), 185–197 (2007).

    Google Scholar 

  6. Chee Y.M., Dau S.H., Ling A.C.H., Ling S.: The sizes of optimal q-ary codes of weight three and distance four: a complete solution. IEEE Trans. Inf. Theory 54(3), 1291–1295 (2008).

    Google Scholar 

  7. Chee Y.M., Dau S.H., Ling A.C.H., Ling S.: Linear size optimal q-ary constant-weight codes and constant-composition codes. IEEE Trans. Inf. Theory 56(1), 140–151 (2010).

    Google Scholar 

  8. Chee Y.M., Ling S.: Constructions for \(q\)-ary constant-weight codes. IEEE Trans. Inf. Theory 53(1), 135–146 (2007).

    Google Scholar 

  9. Chen K., Ge G., Zhu L.: Generalized Steiner triple systems with group size five. J. Comb. Des. 7(6), 441–452 (1999).

    Google Scholar 

  10. Chen K., Ge G., Zhu L.: Starters and related codes. J. Stat. Plan. Inference 86(2), 379–395 (2000).

    Google Scholar 

  11. Colbourn C.J.: Triple systems. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 58–71. CRC Press, Boca Raton (2007).

  12. Colbourn C.J., Kaski P., Östergård P.R.J., Pike D.A., Pottonen O.: Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19. Discret. Math. 311(10–11), 827–834 (2011).

    Google Scholar 

  13. Etzion T.: Optimal constant weight codes over \(Z_k\) and generalized designs. Discret. Math. 169(1), 55–82 (1997).

  14. Fu F.W., Vinck A.J.H., Shen S.Y.: On the constructions of constant-weight codes. IEEE Trans. Inf. Theory 44(1), 328–333 (1998).

    Google Scholar 

  15. Fu F.W., Kløve T., Luo Y., Wei V.K.: On the Svanström bound for ternary constant-weight codes. IEEE Trans. Inf. Theory 47(5), 2061–2064 (2001).

    Google Scholar 

  16. Ge G.: Generalized Steiner triple systems with group size \(g\equiv 1,5\)(mod 6). Australas. J. Comb. 21, 37–47 (2000).

    Google Scholar 

  17. Ge G.: Further results on the existence of generalized Steiner triple systems with group size \(g\equiv 1,5\)(mod 6). Australas. J. Comb. 25, 19–27 (2002).

    Google Scholar 

  18. Ge G.: Generalized Steiner triple systems with group size \(g\equiv 0,3\)(mod 6). Acta Math. Appl. Sin. Engl. Ser. 18(4), 561–568 (2002).

    Google Scholar 

  19. Ge G.: Group divisible designs. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, pp. 255–260. CRC Press, Boca Raton (2007).

  20. Ge G.: Construction of optimal ternary constant weight codes via Bhaskar Rao designs. Discret. Math. 308(13), 2704–2708 (2008).

    Google Scholar 

  21. Ge G., Miao Y.: PBDs, frames, and resolvability. In: Colbourn C.J., Dinitz C.J. (eds.) Handbook of Combinatorial Designs, pp. 261–265. CRC Press, Boca Raton (2007).

  22. Ge G., Wu D.: 4-\(^{*}{\text{ GDDs }}(3^n)\) and generalized Steiner systems GS(2,4,v,3). J. Comb. Des. 11(6), 381–393 (2003).

  23. Ge G., Wu D.: Generalized Steiner triple systems with group size ten. J. Math. Res. Expos. 23(3), 391–396 (2003).

    Google Scholar 

  24. Ge G., Wu D.: Some new optimal quaternary constant weight codes. Sci. China Ser. F 48(2), 192–200 (2005).

    Google Scholar 

  25. Ji L., Wu D., Zhu L.: Existence of generalized Steiner systems GS(2,4, v,2). Des. Codes Cryptogr. 36(1), 83–99 (2005).

    Google Scholar 

  26. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes II. North-Holland Mathematical Library, vol. 6. North-Holland, Amsterdam (1977).

  27. Östergård P.R.J., Svanström M.: Ternary constant weight codes. Electron. J. Comb. 9(1), (2002). Research Paper 41, 23 pp. (electronic).

  28. Phelps K., Yin C.: Generalized Steiner systems with block size three and group size \(g\equiv 3\)(mod 6). J. Comb. Des. 5(6), 417–432 (1997).

    Google Scholar 

  29. Phelps K., Yin C.: Generalized Steiner systems with block size three and group size four. Ars Comb. 53, 133–146 (1999).

    Google Scholar 

  30. Stinson D.R., Wei R., Yin J.: Packings. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 550–556. CRC Press, Boca Raton (2007).

  31. Svanström M.: A lower bound for ternary constant weight codes. IEEE Trans. Inf. Theory 43(5), 1630–1632 (1997).

    Google Scholar 

  32. Svanström M.: Ternary codes with weight constraints. Ph.D. Thesis, Linköpings Universitety, Linköping, Sweden (1999).

  33. Vanstone S.A., Stinson D.R., Schellenberg P.J., Rosa A., Rees R., Colbourn C.J., Cater M.W., Cater J.E.: Hanani triple systems. Isr. J. Math. 83(3), 305–319 (1993).

    Google Scholar 

  34. Wei H., Ge G.: Kirkman frames having hole type \(h^um^1\) for \(h\equiv 0\)(mod 12). Des. Codes Cryptogr. (2012). doi:10.1007/s10623-012-9780-5.

  35. Wu D., Fan P.: Constructions of optimal quaternary constant weight codes via group divisible designs. Discret. Math. 309(20), 6009–6013 (2009).

    Google Scholar 

  36. Wu D., Ge G., Zhu L.: Generalized Steiner triple systems with group size \(g=7,8\). Ars Comb. 57, 175–191 (2000).

    Google Scholar 

  37. Wu D., Zhu L.: Generalized Steiner systems \(\text{ GS } (2,4,\nu ,2)\) with \(\nu \) a prime power \(\equiv 7\)(mod 12). Des. Codes Cryptogr. 24(1), 69–80 (2001).

  38. Yin J., Lu Y., Wang J.: Maximum distance holey packings and related codes. Sci. China Ser. A 42(12), 1262–1269 (1999).

    Google Scholar 

  39. Zhang H., Ge G.: Optimal ternary constant-weight codes of weight four and distance six. IEEE Trans. Inf. Theory 56(5), 2706–2718 (2010).

    Google Scholar 

  40. Zhang H., Ge G.: Optimal quaternary constant-weight codes with weight four and distance five. IEEE Trans. Inf. Theory 59(3), 1617–1629 (2013).

    Google Scholar 

  41. Zhang H., Zhang X., Ge G.: Optimal ternary constant-weight codes with weight 4 and distance 5. IEEE Trans. Inf. Theory 58(5), 2706–2718 (2012).

    Google Scholar 

  42. Zhu M., Ge G.: \(4\)-*GDD\((6^n)\)s and related optimal quaternary constant-weight codes. J. Comb. Des. 20(12), 509–526 (2012).

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Acknowledgments

Research of G. Ge was partially supported by the National Natural Science Foundation of China under Grant No.61171198 and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ13A010001. Research of X. Zhang was partially supported in part by NSFC under Grant No. 11301503.

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Authors and Affiliations

  1. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore

    Yeow Meng Chee, Hui Zhang & Xiande Zhang

  2. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Gennian Ge

  3. Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 100048, China

    Gennian Ge

  4. Department of Mathematics, Zhejiang University, Hangzhou, 310027, Zhejiang, People’s Republic of China

    Hui Zhang

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  1. Yeow Meng Chee
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  2. Gennian Ge
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  4. Xiande Zhang
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Correspondence to Gennian Ge.

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Communicated by T. Etzion.

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Chee, Y.M., Ge, G., Zhang, H. et al. Hanani triple packings and optimal \(q\)-ary codes of constant weight three. Des. Codes Cryptogr. 75, 387–403 (2015). https://doi.org/10.1007/s10623-014-9919-7

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  • DOI: https://doi.org/10.1007/s10623-014-9919-7

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