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Optimal 2-D \((n\times m,3,2,1)\)-optical orthogonal codes and related equi-difference conflict avoiding codes

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Abstract

This paper focuses on constructions for optimal 2-D \((n\times m,3,2,1)\)-optical orthogonal codes with \(m\equiv 0\ (\mathrm{mod}\ 4)\). An upper bound on the size of such codes is established. It relies heavily on the size of optimal equi-difference 1-D (m, 3, 2, 1)-optical orthogonal codes, which is closely related to optimal equi-difference conflict avoiding codes with weight 3. The exact number of codewords of an optimal 2-D \((n\times m,3,2,1)\)-optical orthogonal code is determined for \(n=1,2\), \(m\equiv 0 \pmod {4}\), and \(n\equiv 0 \pmod {3}\), \(m\equiv 8 \pmod {16}\) or \(m\equiv 32 \pmod {64}\) or \(m\equiv 4,20 \pmod {48}\).

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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 (A) 106, 59–75 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  2. Alderson T.L., Mellinger K.E.: \(2\)-Dimensional optical orthogonal codes from Singer groups. Discret. Appl. Math. 157, 3008–3019 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  3. Alderson T.L., Mellinger K.E.: Spreads, arcs, and multiple wavelength codes. Discret. Math. 311, 1187–1196 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  4. Bitan S., Etzion T.: Constructions for optimal constant weight cyclically permutable codes and difference families. IEEE Trans. Inf. Theory 41, 77–87 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Buratti M.: On silver and golden optical orthogonal codes. The Art of Discrete and Applied Mathematics (to appear).

  7. 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 

  8. Buratti M., Momihara K., Pasotti A.: New results on optimal \((v,4,2,1)\) optical orthogonal codes. Des. Codes Cryptogr. 58, 89–109 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  9. Buratti M., Pasotti A., Wu D.: On optimal \((v,5,2,1)\) optical orthogonal codes. Des. Codes Cryptogr. 68, 349–371 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao H., Wei R.: Combinatorial constructions for optimal two-dimensional optical orthogonal codes. IEEE Trans. Inform. Theory 55, 1387–1394 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  11. Chang Y., Miao Y.: Constructions for optimal optical orthogonal codes. Discret. Math. 261, 127–139 (2003).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Chu W., Colbourn C.J.: Recursive constructions for optimal \((n,4,2)\)-OOCs. J. Comb. Des. 12, 333–345 (2004).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Feng T., Chang Y.: Combinatorial constructions for optimal two-dimensional optical orthogonal codes with \(\lambda =2\). IEEE Trans. Inform. Theory 57, 6796–6819 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  16. Feng T., Chang Y., Ji L.: Constructions for strictly cyclic \(3\)-designs and applications to optimal OOCs with \(\lambda =2\). J. Comb. Theory (A) 115, 1527–1551 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  17. Feng T., Wang L., Wang X., Zhao Y.: Optimal two dimensional optical orthogonal codes with the best cross-correlation constraint. J. Comb. Des. 25, 349–380 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  18. Fu H.L., Lin Y.H., Mishima M.: Optimal conflict-avoiding codes of even length and weight \(3\). IEEE Trans. Inform. Theory 56, 5747–5756 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. Fu H.L., Lo Y.H., Shum K.W.: Optimal conflict-avoiding codes of odd length and weight \(3\). Des. Codes Cryptogr. 72, 289–309 (2014).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Ge G., Yin J.: Constructions for optimal \((v,4,1)\) optical orthogonal codes. IEEE Trans. Inform. Theory 47, 2998–3004 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  22. Györfi L., Vajda I.: Constructions of protocol sequences for multiple access collision channel without feedback. IEEE Trans. Inform. Theory 39, 1762–1765 (1993).

    Article  MATH  Google Scholar 

  23. Huang Y., Chang Y.: Two classes of optimal two-dimensional OOCs. Des. Codes Cryptogr. 3, 357–363 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  24. Huang Y., Chang Y.: Maximum two-dimensional \((u\times v,4,1,3)\)-OOCs. Appl. Math. J. Chin. Univ. 28, 279–289 (2013).

    Article  MATH  Google Scholar 

  25. Jimbo M., Mishima M., Janiszewski S., Teymorian A.Y., Tonchev V.D.: On conflict-avoiding codes of length \(n=4m\) for three active users. IEEE Trans. Inform. Theory 53, 2732–2742 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  26. Levenshtein V.I.: Conflict-avoiding codes and cyclic triple systems. Probl. Inf. Transm. 43, 199–212 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  27. Levenshtein V.I., Tonchev V.D.: Optimal conflict-avoiding codes for three active users. In: Proceedings of IEEE International Symposium on Information Theory, Adelaide, Australia, pp. 535–537 (2005).

  28. Lin Y., Mishima M., Satoh J., Jimbo M.: Optimal equi-difference conflict-avoiding codes of odd length and weight three. Finite Fields Appl. 26, 49–68 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  29. Ma W., Zhao C., Shen D.: New optimal constructions of conflict-avoiding codes of odd length and weight \(3\). Des. Codes Cryptogr. 73, 791–804 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  30. Mishima M., Fu H.L., Uruno S.: Optimal conflict-avoiding codes of length \(n\equiv 0 ({\rm mod} 16)\) and weight \(3\). Des. Codes Cryptogr. 52, 275–291 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  31. Miyamoto N., Mizuno H., Shinohara S.: Optical orthogonal codes obtained from conics on finite projective planes. Finite Fields Appl. 10, 405–411 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  32. Momihara K.: Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three. Des. Codes Cryptogr. 45, 379–390 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  33. Momihara K., Buratti M.: Bounds and constructions of optimal \((n,4,2,1)\) optical orthogonal codes. IEEE Trans. Inform. Theory 55, 514–523 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  34. Omrani R., Garg G., Kumar P.V., Elia P., Bhambhani P.: Large families of asymptotically optimal two-dimensional optical orthogonal codes. IEEE Trans. Inform. Theory 58, 1163–1185 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang X., Chang Y.: Further results on \((v,4,1)\)-perfect difference families. Discret. Math. 310, 1995–2006 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang X., Chang Y.: Further results on optimal \((v,4,2,1)\)-OOCs. Discret. Math. 312, 331–340 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang L., Chang Y.: Determination of sizes of optimal three-dimensional optical orthogonal codes of weight three with the AM-OPP restriction. J. Comb. Des. 25, 310–334 (2017).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang J., Shan X., Yin J.: On constructions for optimal two-dimentional optical orthogonal codes. Des. Codes Cryptogr. 54, 43–60 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang X., Chang Y., Feng T.: Optimal \(2\)-D \((n\times m,3,2,1)\)-optical orthogonal codes. IEEE Trans. Inform. Theory 59, 710–725 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  41. Wu S., Fu H.: Optimal tight equi-difference conflict-avoiding codes of length \(n=2k\pm 1\) and weight \(3\). J. Comb. Des. 21, 223–231 (2013).

    Article  MATH  Google Scholar 

  42. Yang G.C., Fuja T.E.: Optical orthogonal codes with unequal auto- and cross-correlation constraints. IEEE Trans. Inform. Theory 41, 96–106 (1995).

  43. Yang G.C., Kwong W.C.: Performance comparison of multiwavelength CDMA and WDMA+CDMA for fiber-optic networks. IEEE Trans. Commun. 45, 1426–1434 (1997).

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors wish to thank the anonymous referees for many valuable comments that greatly improved the quality of this paper; especially, one referee presented Remark 2. This work was supported by NSFC under Grant 11471032, and Fundamental Research Funds for the Central Universities under Grant 2016JBM071, 2016JBZ012 (T. Feng), NSFC under Grant 11401582, NSFHB under Grant A2015507019, and Cultivation Project of National Natural Science Foundation of the Chinese People’s Armed Police Force Academy under Grant ZKJJPY201703 (L. Wang), Zhejiang Provincial Natural Science Foundation of China under Grant LY17A010008, and NSFC under Grant 11771227, 11871291 (X. Wang).

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Tao Feng

  2. Department of Basic Courses, Chinese People’s Armed Police Force Academy, Langfang, 065000, People’s Republic of China

    Lidong Wang

  3. Department of Mathematics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Xiaomiao Wang

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  1. Tao Feng
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  2. Lidong Wang
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  3. Xiaomiao Wang
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Correspondence to Xiaomiao Wang.

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Communicated by M. Buratti.

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Feng, T., Wang, L. & Wang, X. Optimal 2-D \((n\times m,3,2,1)\)-optical orthogonal codes and related equi-difference conflict avoiding codes. Des. Codes Cryptogr. 87, 1499–1520 (2019). https://doi.org/10.1007/s10623-018-0549-3

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