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Compatible difference packing set systems and their applications to multilength variable-weight OOCs

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Abstract

In a study of multilength variable-weight optical orthogonal codes (MLVWOOCs), compatible (N, M, W, 1, Q; 2) difference packing (briefly (N, M, W, 1, Q; 2)-CDP) set systems play an important role. In this paper, a new consequence of Weil’s theorem on multiplicative character sums is presented, some direct constructions of pairwise 2-compatible balanced (n, g, W, 1) difference families (DFs) are obtained for \(W=\{3,4\}\), \(\{3,5\}\), and recursive constructions for (N, M, W, 1, Q; 2)-CDP set systems are derived by means of semicyclic group divisible designs (SCGDDs). Some series of compatible difference packing set systems are produced, and several infinite classes of optimal MLVWOOCs are then obtained.

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References

  1. Abel R.J.R., Buratti M.: Some progress on \((v, 4, 1)\) difference families and optical orthogonal codes. J. Comb. Theory 106, 59–75 (2004).

  2. Bao J., Ji L., Li Y., Wang C.: Orbit-disjoint regular (n, 3, 1)-DPs and their applications to multilength OOCs. Finite Fields Appl. 35, 139–158 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao J., Ji L., Li Y., Wang C.: Some series of optimal multilength OOCs of weight Four. Discret. Math. 338, 2549–2561 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  4. Buratti M.: Cyclic designs with block size 4 and related optimal optical orthogonal codes. Des. Codes Cryptogr. 26, 111–125 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  5. Buratti M.: Constructions of \((q, k, 1)\) difference families with \(q\) a prime power and \(k=4, 5\). Discret. Math. 138, 169–175 (1995).

  6. Buratti M.: Recursive constructions for difference matrices and relative difference families. J. Comb. Des. 6, 165–182 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  7. Buratti M.: Old and new designs via difference multisets and strong difference families. J. Comb. Des. 7, 406–425 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  8. Buratti M., Pasotti A.: Combinatorial designs and the theorem of Weil on multiplicative character sums. Finite Fields Appl. 15, 332–344 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  9. Buratti M., Pasotti A.: Further progress on difference families with block size 4 or 5. Des. Codes Cryptogr. 56, 1–20 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  10. Buratti M., Wei Y., Wu D., Fan P., Cheng M.: Relative difference families with variable block sizes and their related OOCs. IEEE Trans. Inf. Theory 57, 7489–7497 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  11. Chang Y., Fuji-Hara R., Miao Y.: Combinatorial constructions of optimal optical orthogonal codes with weight \(4\). IEEE Trans. Inf. Theory 49, 1283–1292 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen K., Zhu L.: Existence of \((q, 6, 1)\) difference families with \(q\) a prime power. Des. Codes Cryptogr. 15, 167–174 (1998).

  13. Chen K., Wei R., Zhu L.: Existence of \((q, 7, 1)\) difference families with \(q\) a prime power. J. Comb. Des. 10, 126–138 (2002).

  14. Colbourn C.J.: Difference matrices. In: Colbourn C.J., Dinitz J.H. (eds.) CRC Handbook of Combinatorial Designs, pp. 411–419. CRC Press, New York (2007).

    MATH  Google Scholar 

  15. Chung F.R.K., Salehi J.A., Wei V.K.: Optical orthogonal codes: design, analysis, and applications. IEEE Trans. Inf. Theory 35, 595–604 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  16. Chung J.H., Yang K.: New families of optimal variable-weight optical orthogonal codes with high weights. IEEE Trans. Inf. Theory 61, 4511–4517 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  17. Fuji-Hara R., Miao Y.: Optical orthogonal codes: Their bounds and new optimal constructions. IEEE Trans. Inf. Theory 46, 2396–2406 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  18. Kwong W.C., Yang G.C.: Design of multilength optical orthogonal codes for optical CDMA multimedia networks. IEEE Trans. Commun. 50, 1258–1265 (2002).

    Article  Google Scholar 

  19. Lidl R., Neiderreiter H.: Finite Fields, Encyclopedia Math. Cambridge University Press, Cambridge (1983).

    Google Scholar 

  20. Luo X., Yin J., Yue F.: Multilength OOCs: new upper bounds and optimal constructions. IEEE Trans. Inf. Theory 61, 3305–3315 (2015).

    Article  MATH  Google Scholar 

  21. Momihara K.: Strong difference families, difference covers, and their applications for relative difference families. Des. Codes Cryptogr. 51, 253–273 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  22. Maric S.V., Moreno O., Corrada C.J.: Multimedia transmission in fiber-optic LAN’s using optical CDMA. J. Lightwave Technol. 14, 2149–2153 (1996).

    Article  Google Scholar 

  23. Nasaruddin, Tsujioka T: Multiple-length variable-weight optical orthogonal codes for supporting multirate multimedia services in optical CDMA networks. IEICE Trans. Commun. 90, 1968–1978 (2007).

  24. Nasaruddin, Tsujioka T.: OPN02-1: multiple-length variable-weight: optical orthogonal codes for multi-rate multi-quality optical CDMA systems. In: Proc. IEEE Global Telecommun. Conf., San Francisco, CA, USA, Nov./Dec., 1-5 (2006).

  25. Salehi J.A.: Code division multiple access techniques in optical fiber networks-Part I fundamental principles. IEEE Trans. Commun. 37, 824–833 (1989).

    Article  Google Scholar 

  26. Salehi J.A., Brackett C.A.: Code division multiple access techniques in optical fiber networks-Part II systems performance analysis. IEEE Trans. Commun. 37, 834–842 (1989).

    Article  Google Scholar 

  27. Wang C., Yan J., Yin J.: Resolvable generalized difference matrices existence and applications. Finite Fields Appl. 17, 39–56 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang K., Wang J.: Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes. Des. Codes Cryptogr. 63, 305–319 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang J., Yin J.: Two-dimensional optical orthogonal codes and semicyclic group divisible designs. IEEE Trans. Inf. Theory 56, 2177–2187 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  30. Wu D., Zhao H., Fan P., Shinohara S.: Optimal variable-weight optical orthogonal codes via difference packings. IEEE Trans. Inf. Theory 53, 4053–4060 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang G.C.: Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements. IEEE Trans. Commun. 44, 47–55 (1996).

    Article  MATH  Google Scholar 

  32. Yin J.: Some combinatorial constructions for optical orthogonal codes. Discret. Math. 185, 201–219 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  33. Yin J.: A general construction for optimal cyclic differnece packing designs. J. Comb. Theory A 97(2), 272–284 (2002).

    Article  MATH  Google Scholar 

  34. Yin J., Yang X., Li Y.: Some 20-regular DP\((5,1;20u)\) and their applications. Finite Fields Appl. 17, 317–328 (2011).

  35. Zhao H., Qin R.: Bounds and constructions for multilength variable-weight optical orthogonal codes. Discret. Math. 344, 112170 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhao H.: On balanced optimal \((18u, \{3,4\},1)\) optical orthogonal codes. J. Comb. Des. 20, 290–303 (2012).

Download references

Acknowledgements

The authors wish to thank the anonymous referees for their helpful comments and suggestions that much improved the quality of this paper. Research of Rongcun Qin is supported by Guangxi Nature Science Foundation (No. 2019GXNSFBA245021). Research of Hengming Zhao is supported in part by Guangxi Nature Science Foundation (No. 2018GXNSFBA138038) and BAGUI Scholar Program of Guangxi Zhuang Autonomous Region of China (No. 201979). Research of Huangsheng Yu is supported by NSFC (No. 11801103).

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Authors and Affiliations

  1. XingJian College of Science and Liberal Arts, Guangxi University, Nanning, 530005, China

    Rongcun Qin

  2. Department of Mathematics and Information Science, Guangxi College of Education, Nanning, 530023, China

    Hengming Zhao

  3. School of Mathematics and Statistics, Guangxi Normal University, Guilin, 541006, China

    Huangsheng Yu

Authors
  1. Rongcun Qin
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  2. Hengming Zhao
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  3. Huangsheng Yu
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Correspondence to Hengming Zhao.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”

Appendices

Appendix A

Table 2 Two (x, y)s in Lemma 6
Full size table
Table 3 Five \(x'\)s in Lemma 8
Full size table
Table 4 \((x_1,x_2,x_3,x_4,x_5)\) in Lemma 10
Full size table

Appendix B

\({\mathscr {B}}_p\)s for \(p\in \{127,163,181,199,271,307,397,523,919\}\) in Lemma 6.

$$\begin{aligned}&{\mathscr {B}}_{127}: \{(0,0),(0,1),(0,4),(1,16)\}, \{(0,0),(0,16),(1,64),(1,2)\},\\&\quad \{(0,0),(0,64),(1,4)\}, \{(0,0),(0,22),(0,107)\}.\\&{\mathscr {B}}_{163}: \{(0,0),(0,1),(0,149),(1,33)\}, \{(0,0),(0,33),(1,27),(1,111)\},\\&\quad \{(0,0),(0,27),(1,149)\}, \{(0,0),(0,7),(0,17)\}.\\&{\mathscr {B}}_{181}: \{(0,0),(0,1),(0,116),(1,62)\}, \{(0,0),(0,62),(1,133),(1,43)\},\\&\quad \{(0,0),(0,133),(1,116)\}, \{(0,0),(0,10),(0,95)\}.\\&{\mathscr {B}}_{199}: \{(0,0),(0,1),(0,39),(1,128)\}, \{(0,0),(0,128),(1,17),(1,66)\},\\&\quad \{(0,0),(0,17),(1,39)\}, \{(0,0),(0,89),(0,111)\}.\\&{\mathscr {B}}_{271}: \{(0,0),(0,1),(0,38),(1,89)\}, \{(0,0),(0,89),(1,130),(1,62)\},\\&\quad \{(0,0),(0,130),(1,38)\}, \{(0,0),(0,49),(0,35)\}.\\&{\mathscr {B}}_{307}: \{(0,0),(0,1),(0,209),(1,87)\}, \{(0,0),(0,87),(1,70),(1,201)\},\\&\quad \{(0,0),(0,70),(1,209)\}, \{(0,0),(0,59),(0,303)\}.\\&\quad {\mathscr {B}}_{397}: \{(0,0),(0,1),(0,211),(1,57)\}, \{(0,0),(0,57),(1,117),(1,73)\},\\&\quad \{(0,0),(0,117),(1,211)\}, \{(0,0),(0,77),(0,380)\}.\\&{\mathscr {B}}_{523}: \{(0,0),(0,1),(0,377),(1,396)\}, \{(0,0),(0,396),(1,237),(1,439)\},\\&\quad \{(0,0),(0,237),(1,377)\}, \{(0,0),(0,90),(0,461)\}.\\&{\mathscr {B}}_{919}: \{(0,0),(0,1),(0,374),(1,188)\}, \{(0,0),(0,188),(1,468),(1,422)\},\\&\quad \{(0,0),(0,468),(1,374)\}, \{(0,0),(0,3),(0,27)\}. \end{aligned}$$

Appendix C

Base blocks of balanced \(\{3,4\}\)-SCGDDs of type \(u^4\) in Lemma 12.

\(u=6\):

$$\begin{aligned}&\{( 0,0),(1,0),(2,0),(3,0)\}, \{(0,0),( 1,2),(2,1),(3,5)\}, \{(1,0),(2,3),(3,5)\},\\&\quad \{(0,0),(1,4),(3,2)\},\\&\{(0,0),(1,3),(2,5),(3,4)\}, \{(0,0),(1,5),(2,3)\}, \{(0,0),(1,1),(2,2),(3,3)\},\\&\quad \{(0,0),(2,4),(3,1)\}. \end{aligned}$$

\(u=12\): the following base blocks by \((+1 \ (\mathrm{mod\ 4}),-)\).

$$\begin{aligned}&\{(0,0),(1,0),(2,1),(3,3)\}, \{(0,0),(1,3),(2,2),(3,8)\}, \{(0,0),(1,5),(2,0)\},\\&\quad \{(0,0),(1,8),(2,6)\}. \end{aligned}$$

\(u=18\): the following base blocks by \((+1 \ (\mathrm{mod\ 4}),-)\).

$$\begin{aligned}&\{(0,0),(1,6),(2,9),(3,7)\}, \{(0,0),(1,13),(3,6)\}, \{(0,0),(1,2),(2,2),(3,17)\},\\&\{(0,0),(1,17),(3,9)\}, \{(0,0),(1,4),(2,0),(3,8)\}, \{(0,0),(1,5),(2,12)\}. \end{aligned}$$

\(u=24\): the following base blocks by \((+1 \ (\mathrm{mod\ 4}),-)\).

$$\begin{aligned}&\{(0,0),(1,0),(2,23),(3,11)\}, \{(0,0),(1,16),(3,23)\}, \{(0,0),(1,2),(3,2)\},\\&\{(0,0),(1,10),(2,21),(3,4)\}, \{(0,0),(1,3),(2,12),(3,7)\}, \{(0,0),(1,5),(2,2),(3,10)\},\\&\{(0,0),(1,4),(2,10)\}, \{(0,0),(2,15),(3,6)\}. \end{aligned}$$

Appendix D

Base blocks of \((\{3,4\},(\frac{2}{3},\frac{1}{3}))\)-SCGDDs of type \(u^5\) in Example 2.

\(u=6\):

$$\begin{aligned}&\{(0,4),(1,1),(2,3),(3,3)\}, \{(0,3),(1,2),(2,3),(4, 4)\}, \{(0, 3),(2,4),(3,1),(4,1)\},\\&\quad \{(0,0),(1,0),(3,1),(4,3)\}, \{(1,0),(2,4),(3,0),(4,4)\}, \{(0,1),(3,1),(4,0)\},\\&\quad \{(2,3),(3,4),(4,5)\},\\&\quad \{(1,3),(3, 0),(4, 3)\}, \{(1,0),(2,5),(3,4)\}, \{(0,0),(1,4),(3,3)\}, \{(0,0),(2,4),(3,2)\},\\&\quad \{(1,0),(2,3),(4,1)\}, \{(0,5),(1,0),(4,5)\}, \{(0,1),(2,4),(4,3)\}, \{(0,3),(1,5),(2,5)\}. \end{aligned}$$

\(u=12\): the following base blocks by \((+1 \ \mathrm{(mod\ 5)},-)\).

$$\begin{aligned}&\{(0,0),(1,0),(2,2),(3,9)\}, \{(0,0),(1,1),(2,7),(3,11)\}, \{(0,0),(1,8),(3,1)\},\\&\{(0,0),(1,9),(2,0)\}, \{(0,0),(1,5),(2,4)\}, \{(0,0),(2,6),(3,4)\}. \end{aligned}$$

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Qin, R., Zhao, H. & Yu, H. Compatible difference packing set systems and their applications to multilength variable-weight OOCs. Des. Codes Cryptogr. 90, 2613–2645 (2022). https://doi.org/10.1007/s10623-021-00927-y

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