New complementary sets of length 2^m and size 4

Previous Article
Some new two-weight ternary and quinary codes of lengths six and twelve
AMC Home
This Issue
Next Article
New optimal $(v, \{3,5\}, 1, Q)$ optical orthogonal codes

November 2016, 10(4): 825-845. doi: 10.3934/amc.2016043

New complementary sets of length $2^m$ and size 4

Gaofei Wu ^1, , Yuqing Zhang ^2, and Xuefeng Liu ^1,

1.	State Key Laboratory of Integrated Service Networks, Xidian University, Xi'an, 710071, China, China
2.	National Computer Network Intrusion Protection Center, UCAS, Beijing 100043, China

Received January 2015 Revised April 2016 Published November 2016

Abstract
Related Papers

We construct new complementary sequence sets of size $4$, using a graphical description. We explain how the construction can be seen as a special case of a less explicit array construction by Parker and Riera and, at the same time, a generalization of another construction by the same authors. Some generalizations of the construction are also given, which are not in the construction of Parker and Riera. Lower bounds and upper bounds of the number of sequences in the constructions are analyzed.

Keywords: OFDM, Golay sequences, Boolean functions, complementary sets, PMEPR, Reed-Muller codes..

Mathematics Subject Classification: Primary: 94A55, 68P30; Secondary: 05A1.

Citation: Gaofei Wu, Yuqing Zhang, Xuefeng Liu. New complementary sets of length $2^m$ and size 4. Advances in Mathematics of Communications, 2016, 10 (4) : 825-845. doi: 10.3934/amc.2016043

References:

[1]	S. Z. Budĭsin, P. Spasojević, Paraunitary generation/correlation of QAM complementary sequence pairs, Cryptogr. Commun., 6 (2014), 59-102. doi: 10.1007/s12095-013-0087-9.
[2]	C.-Y. Chen, C.-H. Wang, and C.-C. Chao, Complementary sets and Reed-Muller codes for peak-to-average power ratio reduction in OFDM, in Int. Symp. Appl. Algebra Algebr. Algor. Error-Corr. Codes, Springer, Berlin, 2006, 317-327. doi: 10.1007/11617983_31.
[3]	J. A. Davis and J. Jedwab, Peak-to-mean power control for OFDM, Golay complementary sequences, and Reed-Muller codes, IEEE Trans. Inform. Theory, 45 (1999), 2397-2417. doi: 10.1109/18.796380.
[4]	F. Fiedler, J. Jedwab and M. G. Parker, A framework for the construction of Golay sequences, IEEE Trans. Inform. Theory, 54 (2008), 3113-3129. doi: 10.1109/TIT.2008.924667.
[5]	F. Fiedler, J. Jedwab and M. G. Parker, A multi-dimensional approach to the construction and enumeration of Golay complementary sequences, J. Combin. Theory Ser. A, 115 (2008), 753-776. doi: 10.1016/j.jcta.2007.10.001.
[6]	M. J. E. Golay, Complementary series, IEEE Trans. Inform. Theory, 7 (1961), 82-87.
[7]	S. Litsyn, Peak Power Control in Multi-carrier Communications, Cambridge Univ. Press, 2007.
[8]	M. G. Parker and C. Riera, Generalised complementary arrays, IMA Int. Conf. Crypt. Coding, Springer, Berlin, 2011, 41-60. doi: 10.1007/978-3-642-25516-8_4.
[9]	M. G. Parker and C. Tellambura, Golay-Davis-Jedwab complementary sequences and Rudin-Shapiro constructions, in Int. Symp. Inform. Theory, 2001, 302.
[10]	M. G. Parker and C. Tellambura, A construction for binary sequence sets with low peak-to-average power ratio, Reports in Informatics, Univ. Bergen, 2003.
[11]	K. G. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation, IEEE Trans. Inform. Theory, 46 (2000), 104-120. doi: 10.1109/18.817512.
[12]	K.-U. Schmidt, On cosets of the generalized first-order Reed-Muller code with low PMEPR, IEEE Trans. Inform. Theory, 52 (2006), 3220-3232. doi: 10.1109/TIT.2006.876252.
[13]	K.-U. Schmidt, Complementary sets, generalized Reed-Muller codes, and power control for OFDM, IEEE Trans. Inform. Theory, 53 (2007), 808-814. doi: 10.1109/TIT.2006.889723.
[14]	T. E. Stinchcombe, Aperiodic autocorrelations of length $2^m$ sequences, complementarity, and power control for OFDM, Ph.D. thesis, Univ. London, 2000.
[15]	Z. Wang, M. G. Parker, G. Gong and G. Wu, On the PMEPR of the binary Golay sequences of length $2^n$, IEEE Trans. Inform. Theory, 60 (2014), 2391-2398. doi: 10.1109/TIT.2014.2300867.
[16]	Z. Wang, G. Wu and H. Li, Improved PMEPR bound on near-complementary sequences constructed by Yu and Gong, Electr. Lett., 49 (2013), 73-75.
[17]	G. Wu and M. G. Parker, A complementary construction using mutually unbiased bases, Crypt. Commun., 6 (2014), 3-25. doi: 10.1007/s12095-013-0095-9.
[18]	G. Wu, Y. Zhang and Z. Wang, Construction of near-complementary sequences with low PMEPR for peak power control in OFDM, IEICE Trans. Fundam., E95-A (2012), 1881-1887.
[19]	N. Y. Yu and G. Gong, Near-complementary sequences with low PMEPR for peak power control in multicarrier communications, IEEE Trans. Inform. Theory, 57 (2011), 505-513. doi: 10.1109/TIT.2010.2090230.

show all references

References:

[1]	S. Z. Budĭsin, P. Spasojević, Paraunitary generation/correlation of QAM complementary sequence pairs, Cryptogr. Commun., 6 (2014), 59-102. doi: 10.1007/s12095-013-0087-9.
[2]	C.-Y. Chen, C.-H. Wang, and C.-C. Chao, Complementary sets and Reed-Muller codes for peak-to-average power ratio reduction in OFDM, in Int. Symp. Appl. Algebra Algebr. Algor. Error-Corr. Codes, Springer, Berlin, 2006, 317-327. doi: 10.1007/11617983_31.
[3]	J. A. Davis and J. Jedwab, Peak-to-mean power control for OFDM, Golay complementary sequences, and Reed-Muller codes, IEEE Trans. Inform. Theory, 45 (1999), 2397-2417. doi: 10.1109/18.796380.
[4]	F. Fiedler, J. Jedwab and M. G. Parker, A framework for the construction of Golay sequences, IEEE Trans. Inform. Theory, 54 (2008), 3113-3129. doi: 10.1109/TIT.2008.924667.
[5]	F. Fiedler, J. Jedwab and M. G. Parker, A multi-dimensional approach to the construction and enumeration of Golay complementary sequences, J. Combin. Theory Ser. A, 115 (2008), 753-776. doi: 10.1016/j.jcta.2007.10.001.
[6]	M. J. E. Golay, Complementary series, IEEE Trans. Inform. Theory, 7 (1961), 82-87.
[7]	S. Litsyn, Peak Power Control in Multi-carrier Communications, Cambridge Univ. Press, 2007.
[8]	M. G. Parker and C. Riera, Generalised complementary arrays, IMA Int. Conf. Crypt. Coding, Springer, Berlin, 2011, 41-60. doi: 10.1007/978-3-642-25516-8_4.
[9]	M. G. Parker and C. Tellambura, Golay-Davis-Jedwab complementary sequences and Rudin-Shapiro constructions, in Int. Symp. Inform. Theory, 2001, 302.
[10]	M. G. Parker and C. Tellambura, A construction for binary sequence sets with low peak-to-average power ratio, Reports in Informatics, Univ. Bergen, 2003.
[11]	K. G. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation, IEEE Trans. Inform. Theory, 46 (2000), 104-120. doi: 10.1109/18.817512.
[12]	K.-U. Schmidt, On cosets of the generalized first-order Reed-Muller code with low PMEPR, IEEE Trans. Inform. Theory, 52 (2006), 3220-3232. doi: 10.1109/TIT.2006.876252.
[13]	K.-U. Schmidt, Complementary sets, generalized Reed-Muller codes, and power control for OFDM, IEEE Trans. Inform. Theory, 53 (2007), 808-814. doi: 10.1109/TIT.2006.889723.
[14]	T. E. Stinchcombe, Aperiodic autocorrelations of length $2^m$ sequences, complementarity, and power control for OFDM, Ph.D. thesis, Univ. London, 2000.
[15]	Z. Wang, M. G. Parker, G. Gong and G. Wu, On the PMEPR of the binary Golay sequences of length $2^n$, IEEE Trans. Inform. Theory, 60 (2014), 2391-2398. doi: 10.1109/TIT.2014.2300867.
[16]	Z. Wang, G. Wu and H. Li, Improved PMEPR bound on near-complementary sequences constructed by Yu and Gong, Electr. Lett., 49 (2013), 73-75.
[17]	G. Wu and M. G. Parker, A complementary construction using mutually unbiased bases, Crypt. Commun., 6 (2014), 3-25. doi: 10.1007/s12095-013-0095-9.
[18]	G. Wu, Y. Zhang and Z. Wang, Construction of near-complementary sequences with low PMEPR for peak power control in OFDM, IEICE Trans. Fundam., E95-A (2012), 1881-1887.
[19]	N. Y. Yu and G. Gong, Near-complementary sequences with low PMEPR for peak power control in multicarrier communications, IEEE Trans. Inform. Theory, 57 (2011), 505-513. doi: 10.1109/TIT.2010.2090230.

[1]	Martino Borello, Olivier Mila. Symmetries of weight enumerators and applications to Reed-Muller codes. Advances in Mathematics of Communications, 2019, 13 (2) : 313-328. doi: 10.3934/amc.2019021
[2]	Daniele Bartoli, Adnen Sboui, Leo Storme. Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes. Advances in Mathematics of Communications, 2016, 10 (2) : 355-365. doi: 10.3934/amc.2016010
[3]	Olav Geil, Stefano Martin. Relative generalized Hamming weights of q-ary Reed-Muller codes. Advances in Mathematics of Communications, 2017, 11 (3) : 503-531. doi: 10.3934/amc.2017041
[4]	Andreas Klein, Leo Storme. On the non-minimality of the largest weight codewords in the binary Reed-Muller codes. Advances in Mathematics of Communications, 2011, 5 (2) : 333-337. doi: 10.3934/amc.2011.5.333
[5]	Bingsheng Shen, Yang Yang, Ruibin Ren. Three constructions of Golay complementary array sets. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022019
[6]	Tinghua Hu, Yang Yang, Zhengchun Zhou. Golay complementary sets with large zero odd-periodic correlation zones. Advances in Mathematics of Communications, 2021, 15 (1) : 23-33. doi: 10.3934/amc.2020040
[7]	Frank Fiedler. Small Golay sequences. Advances in Mathematics of Communications, 2013, 7 (4) : 379-407. doi: 10.3934/amc.2013.7.379
[8]	Yang Yang, Xiaohu Tang, Guang Gong. Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 113-125. doi: 10.3934/amc.2013.7.113
[9]	Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543
[10]	Constanza Riera, Pantelimon Stănică. Landscape Boolean functions. Advances in Mathematics of Communications, 2019, 13 (4) : 613-627. doi: 10.3934/amc.2019038
[11]	Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[12]	B. K. Dass, Namita Sharma, Rashmi Verma. Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 629-639. doi: 10.3934/amc.2018037
[13]	Zilong Wang, Guang Gong, Rongquan Feng. A generalized construction of OFDM $M$-QAM sequences with low peak-to-average power ratio. Advances in Mathematics of Communications, 2009, 3 (4) : 421-428. doi: 10.3934/amc.2009.3.421
[14]	Yujuan Li, Guizhen Zhu. On the error distance of extended Reed-Solomon codes. Advances in Mathematics of Communications, 2016, 10 (2) : 413-427. doi: 10.3934/amc.2016015
[15]	Peter Beelen, David Glynn, Tom Høholdt, Krishna Kaipa. Counting generalized Reed-Solomon codes. Advances in Mathematics of Communications, 2017, 11 (4) : 777-790. doi: 10.3934/amc.2017057
[16]	Antonio Cafure, Guillermo Matera, Melina Privitelli. Singularities of symmetric hypersurfaces and Reed-Solomon codes. Advances in Mathematics of Communications, 2012, 6 (1) : 69-94. doi: 10.3934/amc.2012.6.69
[17]	Sihem Mesnager, Gérard Cohen. Fast algebraic immunity of Boolean functions. Advances in Mathematics of Communications, 2017, 11 (2) : 373-377. doi: 10.3934/amc.2017031
[18]	Claude Carlet, Serge Feukoua. Three basic questions on Boolean functions. Advances in Mathematics of Communications, 2017, 11 (4) : 837-855. doi: 10.3934/amc.2017061
[19]	Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069
[20]	Claude Carlet. Parameterization of Boolean functions by vectorial functions and associated constructions. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022013

2020 Impact Factor: 0.935

Tools

Metrics

Other articles
by authors

on AIMS
Gaofei Wu Yuqing Zhang Xuefeng Liu
on Google Scholar
Gaofei Wu Yuqing Zhang Xuefeng Liu

[Back to Top]