Optimal three-dimensional optical orthogonal codes of weight three. (English) Zbl 1311.05029
Optical orthogonal codes (OOCs) can be considered in the framework of packing designs with an automorphism that acts cyclically on the points. One may generalize OOCs to the \(d\)-dimensional case, where the codewords are \(d\)-dimensional 0-1 matrices. Results on 2-dimensional OOCs are presented, for example, by R. Omrani et al. [“Large families of optimal two-dimensional optical orthogonal codes”, IEEE Trans. Inform. Theory 58, No. 2, 1163–1185 (2012; doi:10.1109/TIT.2011.2169299)]. S. Kim et. al. [“A new family of space/wavelength/time spread three-dimensional optical code for OCDMA network”, J. Lightwave Technol. 18, No. 4, 502–511 (2000; doi:10.1109/50.838124)] give a motivation for the 3-dimensional case.
The author studies 3-dimensional OOCs. In addition to the basic auto-correlation and cross-correlation requirements, some further conditions are assumed here, leading to the study of some subclasses of 3-dimensional OOCs. These subclasses are called SPPC (single-pulse-per-plane code) and AMOPPC (at-most-one-pulse-per-plane code). For certain parameters, the structures are equivalent to generalized Bhaskar Rao group divisible designs, signed over a cyclic group. Particular emphasis is put on the case of weight 3 (when there are exactly three 1s in the 3-dimensional 0-1 matrices).
The author studies 3-dimensional OOCs. In addition to the basic auto-correlation and cross-correlation requirements, some further conditions are assumed here, leading to the study of some subclasses of 3-dimensional OOCs. These subclasses are called SPPC (single-pulse-per-plane code) and AMOPPC (at-most-one-pulse-per-plane code). For certain parameters, the structures are equivalent to generalized Bhaskar Rao group divisible designs, signed over a cyclic group. Particular emphasis is put on the case of weight 3 (when there are exactly three 1s in the 3-dimensional 0-1 matrices).
Reviewer: Patric R. J. Östergård (Helsinki)
References:
| [1] | Abel R.J.R., Buratti M.: Some progress on \[(v,\,4,\,1)\](v,4,1) difference families and optical orthogonal codes. J. Comb. Theory A 106(1), 59-75 (2004). · Zbl 1061.94068 |
| [2] | Abel R.J.R., Chan N.H.N., Combe D., Palmer W.D.: Existence of GBRDs with block size 4 and BRDs with block size 5. Des. Codes Cryptogr. 61(3), 285-300 (2011). · Zbl 1229.05045 |
| [3] | Beth T., Jungnickel D., Lenz H.: Design theory. In: Encyclopedia of Mathematics and Its Applications, vol. 78, 2nd edn. Cambridge University Press, Cambridge (1999). · Zbl 0945.05004 |
| [4] | Blake-Wilson S., Phelps K.T.: Constant weight codes and group divisible designs. Des. Codes Cryptogr. 16, 11-27 (1999). · Zbl 0945.05011 |
| [5] | Buratti M.: Cyclic designs with block size 4 and related optimal optical orthogonal codes. Des. Codes Cryptogr. 26, 111-125 (2002). · Zbl 1003.05016 |
| [6] | Buratti M., Pasotti A.: Further progress on difference families with block size 4 or 5. Des. Codes Cryptogr. 56(1), 1-20 (2010). · Zbl 1198.94186 |
| [7] | Chang Y., Fuji-Hara R., Miao Y.: Combinatorial constructions of optimal optical orthogonal codes with weight 4. IEEE Trans. Inf. Theory 49(5), 1283-1292 (2003). · Zbl 1063.94053 |
| [8] | Chang Y., Yin J.: Further results on optimal optical orthogonal codes with weight 4. Discret. Math. 279(1-3), 135-151 (2004). · Zbl 1046.94015 |
| [9] | Chee Y.M., Ge G., Ling A.C.H.: Group divisible codes and their application in the construction of optimal constant-composition codes of weight three. IEEE Trans. Inf. Theory 54(8), 3552-3564 (2008). · Zbl 1181.94123 |
| [10] | Chen K., Zhu L.: Existence of \[(q,\,6,\,1)\](q,6,1) difference families with \[q\] q a prime power. Des. Codes Cryptogr. 15(2), 167-173 (1998). · Zbl 0912.05013 |
| [11] | 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). · Zbl 0676.94021 |
| [12] | Colbourn C.J., Rosa A.: Triple Systems. Oxford Mathematics Monographs. Oxford University Press, Oxford (1999). · Zbl 0938.05009 |
| [13] | Dai P., Wang J., Yin J.: Combinatorial constructions for optimal 2-D optical orthogonal codes with AM-OPPTS property. Des. Codes Cryptogr. 1-16 (2012). doi: 10.1007/s10623-012-9733-z. · Zbl 1286.05025 |
| [14] | de Launey W.: Bhaskar Rao designs. In: CRC Handbook of Combinatorial Designs, pp. 299-301. CRC Press, Boca Raton (2007). |
| [15] | Djordjevic I.B., Vasic B.: Novel combinatorial constructions of optical orthogonal codes for incoherent optical CDMA systems. J. Lightwave Technol. 21(9), 1869-1875 (2003). |
| [16] | Etzion T.: Optimal constant weight codes over \[Z_k\] Zk and generalized designs. Discret. Math. 169, 55-82 (1997). · Zbl 0974.05016 |
| [17] | Feng T., Chang Y.: Combinatorial constructions for optimal two-dimensional optical orthogonal codes with \[\lambda = 2\] λ=2. IEEE Trans. Inf. Theory 57(10), 6796-6819 (2011). · Zbl 1365.94556 |
| [18] | Fuji-Hara R., Miao Y.: Optical orthogonal codes: their bounds and new optimal constructions. IEEE Trans. Inf. Theory 46(11), 2396-2406 (2000). · Zbl 1009.94014 |
| [19] | Gallant R.P., Jiang Z., Ling A.C.H.: The spectrum of cyclic group divisible designs with block size three. J. Comb. Des. 7, 95-105 (1999). · Zbl 0931.05010 |
| [20] | Gao F., Ge G.: Optimal ternary constant-composition codes of weight four and distance five. IEEE Trans. Inf. Theory 57(6), 3742-3757 (2011). · Zbl 1365.94571 |
| [21] | Garg M.: Performance analysis and implementation of three dimensional codes in optical code division multiple access system. Master, Thapar University, Punjab (2012). |
| [22] | Ge G.: On \[(g,\,4;\,1)\](g,4;1)-difference matrices. Discret. Math. 301, 164-174 (2005). · Zbl 1081.05015 |
| [23] | Ge G.: Construction of optimal ternary constant weight codes via Bhaskar Rao designs. Discret. Math. 308(13), 2704-2708 (2008). · Zbl 1158.05009 |
| [24] | Ge G., Greig M., Seberry J.: Generalized Bhaskar Rao designs with block size 4 signed over elementary Abelian groups. J. Comb. Math. Comb. Comput. 46, 3-45 (2003). · Zbl 1037.05007 |
| [25] | Ge G., Greig M., Seberry J., Seberry, R.: Generalized Bhaskar Rao designs with block size 3 over finite Abelian groups. Graphs Comb. 23(3), 271-290 (2007). · Zbl 1122.05011 |
| [26] | Hanani H.: Balanced incomplete block designs and related designs. Discret. Math. 11, 255-369 (1975). · Zbl 0361.62067 |
| [27] | Jiang Z.: Concerning cyclic group divisible designs with block size three. Australas. J. Comb. 13, 227-245 (1996). · Zbl 0847.05013 |
| [28] | Jindal S., Gupta N.: Performance evaluation of optical CDMA based 3D code with increasing bit rate in local area network. In: IEEE Region 8 Computational Technologies in Electrical and Electronics Engineering (SIBIRCON), Novosibirsk, Russia, pp. 386-388, 2008. |
| [29] | Kim S., Yu K., Park N.: A new family of space/wavelength/time spread three-dimensional optical code for OCDMA network. J. Lightwave Technol. 18(4), 502-511 (2000). |
| [30] | Li X., Fan P., Shum K.W.: Construction of space/wavelength/time spread optical code with large family size. IEEE Commun. Lett. 16(6), 893-896 (2012). |
| [31] | Li X., Fan P., Suehiro N., Wu D.: New classes of optimal variable-weight optical orthogonal codes with Hamming weights 3 and 4. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E95-A(11), 1843-1850 (2012). |
| [32] | McGeehan J.E., Motaghian Nezam S.M.R., Saghari P., Willner A.E., Omrani R., Kumar P.V.: Experimental demonstration of OCDMA transmission using a three-dimensional (time-wavelength-polarization) codeset. J. Lightwave Technol. 23(10), 3283-3289 (2005). |
| [33] | Miyamoto N., Mizuno H., Shinohara S.: Optical orthogonal codes obtained from conics on finite projective planes. Finite Fields Appl. 10(3), 405-411 (2004). · Zbl 1046.94016 |
| [34] | Omrani R., Kumar P.V., Elia P., Bhambhani P.: Large families of optimal two-dimensional optical orthogonal codes. IEEE Trans. Inf. Theory 58(2), 1163-1185 (2012). · Zbl 1365.94648 |
| [35] | Ortiz-Ubarri J., Moreno O., Tirkel A.: Three-dimensional periodic optical orthogonal code for OCDMA systems. In: IEEE Information Theory Workshop (ITW), Paraty, Brazil, pp. 170-174, 2011. |
| [36] | Rotman J.J.: Advanced Modern Algebra. Graduate Studies in Mathematics, 2nd edn. American Mathematic Society, Providence (2009). · Zbl 0997.00001 |
| [37] | Sawa M.: Optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4. IEEE Trans. Inf. Theory 56(7), 3613-3620 (2010). · Zbl 1366.94613 |
| [38] | Wang J., Shan X., Yin J.: On constructions for optimal two-dimensional optical orthogonal codes. Des. Codes Cryptogr. 54, 43-60 (2010). · Zbl 1230.05092 |
| [39] | Wang J., Yin J.: Two-dimensional optimal optical orthogonal codes and semicyclic group divisible designs. IEEE Trans. Inf. Theory 56(5), 2177-2187 (2010). · Zbl 1366.94673 |
| [40] | Wang K., Wang J.: Semicyclic 4-GDDs and related two-dimensional optical orthogonal codes. Des. Codes Cryptogr. 63(3), 305-319 (2012). · Zbl 1238.05031 |
| [41] | Wang X., Chang Y., Feng T.: Optimal 2-D \[(n\times m\] n×m, 3, 2, 1)-optical orthogonal codes. IEEE Trans. Inf. Theory 59(1), 710-725 (2013). · Zbl 1364.94688 |
| [42] | Wilson R.M.: An existence theory for pairwise balanced designs. J. Comb. Theory A 13, 220-245 (1972). · Zbl 0263.05014 |
| [43] | Yin J.: Some combinatorial constructions for optical orthogonal codes. Discret. Math. 185, 201-219 (1998). · Zbl 0949.94010 |
| [44] | 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). · Zbl 1365.94613 |
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