Balanced ($\mathbb{Z} _{2u}\times \mathbb{Z}_{38v}$, {3, 4, 5}, 1) difference packings and related codes

Hengming Zhao; Rongcun Qin; Dianhua Wu

doi:10.3934/amc.2022008

Article Contents

2024, Volume 18, Issue 1: 163-178. Doi: 10.3934/amc.2022008

This issue Previous Article Computing square roots faster than the Tonelli-Shanks/Bernstein algorithm Next Article The zero-error capacity of binary channels with 2-memories

Balanced ($\mathbb{Z} _{2u}\times \mathbb{Z}_{38v}$, {3, 4, 5}, 1) difference packings and related codes

1.
Department of Mathematics and Information Science, Guangxi College of Education, Nanning 530023, China
2.
Department of Mathematics, Guangxi Agricultural Vocational University, Nanning 530007, China
3.
School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

^*Corresponding author: Dianhua Wu
^*Corresponding author: Dianhua Wu

Received: September 2021

Revised: January 2022

Early access: February 2022

Published: February 2024

The first author is supported by BAGUI Scholar Program of Guangxi Zhuang Autonomous Region of China (No. 201979). The second author is supported in part by Guangxi Nature Science Foundation (No. 2019GXNSFBA245021), and the Project of Basic Ability Improvement of Young and Middle-aged Teachers of Universities in Guangxi (No. 2020KY54014). The last author is supported in part by NSFC (No. 12161010, 11801103)

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Abstract

Let $ m $, $ n $ be positive integers, and $ K $ a set of positive integers with size greater than 2. An $ (m,n,K,1) $ optical orthogonal signature pattern code, $ (m,n,K,1) $-OOSPC, was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirement. Let $ G $ be an additive group, a balanced $ (G, K, 1) $ difference packing, $ (G, K, 1) $-BDP, can be used to construct a balanced $ (m,n,K,1) $-OOSPC when $ G = {\mathbb{Z}}_m\times {\mathbb{Z}}_n $. In this paper, the existences of optimal $ ( {\mathbb{Z}}_{2u}\times {\mathbb{Z}}_{38v}, \{3,4,5\},1) $-BDPs are completely solved with $ u, \ v\equiv 1\pmod2 $, and the corresponding optimal balanced $ (2u, 38v,\{3,4,5\},1) $-OOSPCs are also obtained.

Keywords:

Balanced,
difference matrix,
difference packing,
optical orthogonal signature pattern code,
semi-cyclic group divisible design.

Mathematics Subject Classification: Primary: 05B40; Secondary: 94C30.

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[1]	R. J. R. Abel, C. J. Colbourn and J. H. Dinitz, Mutually orthogonal latin squares (MOLS), Handbook of Combinatorial Designs, New York: CRC Press, (2007), 160–193.
[2]	S. Bonvicini, M. Buratti, M. Garonzi and G. Traetta, The first families of highly symmetric Kirkman triple systems whose orders fill a congruence class, Designs, Codes and Cryptography, 89 (2021), 2725-2757. doi: 10.1007/s10623-021-00952-x.
[3]	M. Buratti, A power method for constructing difference families and optimal optical orthogonal codes, Des. Codes Cryptogr., 5 (1995), 13-25. doi: 10.1007/BF01388501.
[4]	M. Buratti, Hadamard partitioned difference families and their descendants, Cryptogr. Commun., 11 (2019), 557-562. doi: 10.1007/s12095-018-0308-3.
[5]	M. Buratti, Old and new designs via difference multisets and strong difference families, J. Combin. Des., 7 (1999), 406-425. doi: 10.1002/(SICI)1520-6610(1999)7:6<406::AID-JCD2>3.0.CO;2-U.
[6]	M. Buratti, On silver and golden optical orthogonal codes, Art Discrete Appl. Math., 1 (2018), Paper No. 2.02, 11 pp. doi: 10.26493/2590-9770.1236. ce4.
[7]	M. Buratti, Recursive constructions for difference matrices and relative difference families, J. Combin. Des., 6 (1998), 165-182. doi: 10.1002/(SICI)1520-6610(1998)6:3<165::AID-JCD1>3.0.CO;2-D.
[8]	M. Buratti, Y. Wei, D. Wu, P. Fan and M. Cheng, Relative difference families with variable block sizes and their related OOCs, IEEE Trans. Inform. Theory, 57 (2011), 7489-7497. doi: 10.1109/TIT.2011.2162225.
[9]	M. Buratti, J. Yan and C. Wang, From a 1-rotational RBIBD to a partitioned difference family, Electron. J. Combin., 17 (2010), 139, 23 pp.
[10]	M. Buratti and F. Zuanni, G-Invariantly resolvable Steiner 2-designs arising from 1-rotational difference families, Bull. Belg. Math. Soc., 5 (1998), 221-235.
[11]	Y. Chang, S. Costa, T. Feng and X. Wang, Strong difference families of special types, Discr. Math., 343 (2020), 111776, 12 pp. doi: 10.1016/j. disc. 2019.111776.
[12]	K. Chen, G. Ge and L. Zhu, Starters and related codes, J. Statist. Plann. Inference, 86 (2000), 379-395. doi: 10.1016/S0378-3758(99)00119-6.
[13]	J. Chen, L. Ji and Y. Li, Combinatorial constructions of optimal $(m, n, 4, 2)$ optical orthogonal signature pattern codes, Des. Codes Cryptogr., 86 (2018), 1499-1525. doi: 10.1007/s10623-017-0409-6.
[14]	J. Chen, L. Ji and Y. Li, New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, Des. Codes Cryptogr., 85 (2017), 299-318. doi: 10.1007/s10623-016-0310-8.
[15]	C. J. Colbourn, Difference matrices, In: C. J. Colbourn, J. H. Dinitz, eds. CRC Handbook of Combinatorial Designs, New York: CRC Press, (2007), 411–419.
[16]	S. Costa, T. Feng and X. Wang, Frame difference families and resolvable balanced incomplete block designs, Des. Codes Cryptogr., 86 (2018), 2725-2745. doi: 10.1007/s10623-018-0472-7.
[17]	S. Costa, T. Feng and X. Wang, New 2-designs from strong difference families, Finite Fields Appl., 50 (2018), 391-405. doi: 10.1016/j.ffa.2017.12.011.
[18]	I. B. Djordjevic, B. Vasic and J. Rorison, Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs, J. Lightw. Technol., 21 (2003), 1850-1856. doi: 10.1109/JLT.2003.816819.
[19]	G. Ge, On $(g, 4;1)$-diffference matrices, Discr. Math., 301 (2005), 164-174. doi: 10.1016/j.disc.2005.07.004.
[20]	G. Ge and J. Yin, Constructions for optimal $(v, 4, 1)$ optical orthogonal codes, IEEE Trans. Inform. Theory, 47 (2001), 2998-3004. doi: 10.1109/18.959278.
[21]	F. R. Gu and J. Wu, Construction and performance analysis of variable-weight optical orthogonal codes for asynchronous optical CDMA systems, J. Lightw. Technol., 23 (2005), 740-748. doi: 10.1109/JLT.2004.838880.
[22]	L. Ji, B. Ding, X. Wang and G. Ge, Asymptotically optimal optical orthogonal signature pattern codes, IEEE Trans. Inform. Theory, 64 (2018), 5419-5431. doi: 10.1109/TIT.2017.2787593.
[23]	J. Jiang, D. Wu and M. H. Lee, Some infinte classes of optimal $(v, \{3, 4\}, 1, Q)$-OOCs with $Q\in\{\{1/3, 2/3\}, \{2/3, 1/3\}\}$, Graphs Combin., 29 (2013), 1795-1811. doi: 10.1007/s00373-012-1235-2.
[24]	K. Kitayama, Novel spatial spread spectrum based fiber optic CDMA networks for image transmission, IEEE J. Select. Areas Commun., 12 (1994), 762-772. doi: 10.1109/49.286683.
[25]	W. C. Kwong and G. C. Yang, Double-weight signature pattern codes for multicore-fiber code-division multiple-access networks, IEEE Commun. Lett., 5 (2001), 203-205. doi: 10.1109/4234.922760.
[26]	W. Li, H. Zhao, R. Qin and D. Wu, Constructions of optimal balanced $(m, n, \{4, 5\}, 1)$-OOSPCs, Adv. Math. Commun., 13 (2019), 329-341. doi: 10.3934/amc.2019022.
[27]	S. Ma and Y. Chang, A new class of optimal optical orthogonal codes with weight five, IEEE Trans. Inform. Theory, 50 (2004), 1848-1850. doi: 10.1109/TIT.2004.831845.
[28]	K. Momihara, Strong difference families, difference covers, and their applications for relative difference families, Des. Codes Cryptogr., 51 (2009), 253-273. doi: 10.1007/s10623-008-9259-6.
[29]	R. Pan and Y. Chang, Combinatorial constructions for maximum optical orthogonal signature pattern codes, Discr. Math., 313 (2013), 2918-2931. doi: 10.1016/j.disc.2013.09.005.
[30]	R. Pan and Y. Chang, Determination of the sizes of optimal $(m, n, k, \lambda, k-1)$-OOSPCs for $\lambda=k-1, k$, Discr. Math., 313 (2013), 1327-1337. doi: 10.1016/j.disc.2013.02.019.
[31]	R. Pan and Y. Chang, Further results on optimal $(m, n, 4, 1)$ optical orthogonal signature pattern codes (in Chinese), Sci. Sin. Math., 44 (2014), 1141-1152.
[32]	R. Pan and Y. Chang, $(m, n, 3, 1)$ optical orthogonal signature pattern codes with maximum possible size, IEEE Trans. Inform. Theory, 61 (2015), 1139-1148. doi: 10.1109/TIT.2014.2381259.
[33]	R. Qin and H. Zhao, Constructions for optimal $(u\times v, \{3, 4\}, 1, Q)$-OOSPCs, Discr. Math., 342 (2019), 1924-1948. doi: 10.1016/j.disc.2019.03.006.
[34]	J. A. Salehi, Emerging optical code-division multiple access communications systems, IEEE Network, 3 (1989), 31-39. doi: 10.1109/65.21908.
[35]	M. Sawa, Optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4, IEEE Trans. Inform. Theory, 56 (2010), 3613-3620. doi: 10.1109/TIT.2010.2048487.
[36]	M. Sawa and S. Kageyama, Optimal optical orthogonal signature pattern codes of weight 3, Biom. Lett., 46 (2009), 89-102.
[37]	X. Wang, T. Feng and S. Wang, Existence of strong difference families and constructions for eight new $2$-designs, J. Combin. Des., 29 (2021), 225-242. doi: 10.1002/jcd.21765.
[38]	D. Wu, H. Zhao, P. Fan and S. Shinohara, Optimal variable-weight optical orthogonal codes via difference packings, IEEE Trans. Inform. Theory, 56 (2010), 4053-4060. doi: 10.1109/TIT.2010.2050927.
[39]	G. C. Yang, Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements, IEEE Trans. Commun., 44 (1996), 47-55.
[40]	G. C. Yang and W. C. Kwong, Two-dimensional spatial signature patterns, IEEE Trans. Commun., 44 (1996), 184-191.
[41]	J. Yin, A general construction for optimal cyclic packing designs, J. Combin. Theory (Ser. A), 97 (2002), 272-284. doi: 10.1006/jcta.2001.3215.
[42]	J. Yin, Some combinatorial constructions for optical orthogonal codes, Discr. Math., 185 (1998), 201-219. doi: 10.1016/S0012-365X(97)00172-6.
[43]	H. Zhao, On balanced optimal $(18u, \{3, 4\}, 1)$ optical orthogonal codes, J. Combin. Des., 20 (2012), 290-303. doi: 10.1002/jcd.21303.
[44]	H. Zhao and R. Qin, Combinatorial constructions for optimal multiple-weight optical orthogonal signature pattern codes, Discr. Math., 339 (2016), 179-193. doi: 10.1016/j.disc.2015.08.005.
[45]	H. Zhao, R. Qin and D. Wu, On balanced $({\mathbb{Z}}_4u\times {\mathbb{Z}}_8v, \{4, 5\}, 1)$ difference packings, Discr. Math., 344 (2021), 112552, 13 pp. doi: 10.1016/j. disc. 2021.112552.
[46]	H. Zhao, D. Wu and P. Fan, Constructions of optimal variable-weight optical orthogonal codes, J. Combin. Des., 18 (2010), 274-291. doi: 10.1002/jcd.20246.