nach oben

Designs, Codes and Cryptography

Erschienen in:

01.06.2015

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

verfasst von: Yeow Meng Chee, Gennian Ge, Hui Zhang, Xiande Zhang

Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2015

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel New results on nonexistence of generalized bent functions

Nächster Artikel Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

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

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

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

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).

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

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).

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).

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

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

10.

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

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).

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).

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).

16.

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

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).

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).

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).

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).

24.

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

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).

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).

29.

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

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).

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).

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).

36.

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

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).

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).

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).

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).

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).

Titel: Hanani triple packings and optimal -ary codes of constant weight three
verfasst von: Yeow Meng Chee
Gennian Ge
Hui Zhang
Xiande Zhang
Publikationsdatum: 01.06.2015
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 3/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-014-9919-7

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 3/2015

False positive probabilities in q-ary Tardos codes: comparison of attacks

New results on nonexistence of generalized bent functions

The largest Erdős–Ko–Rado sets in designs

Fourier-reflexive partitions and MacWilliams identities for additive codes

Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

New cube root algorithm based on the third order linear recurrence relations in finite fields