# American Institute of Mathematical Sciences

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

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

 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

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.
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:
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##### References:
 [1] S. Z. Budĭsin, P. Spasojević, Paraunitary generation/correlation of QAM complementary sequence pairs,, Cryptogr. Commun., 6 (2014), 59.  doi: 10.1007/s12095-013-0087-9.  Google Scholar [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, (2006), 317.  doi: 10.1007/11617983_31.  Google Scholar [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.  doi: 10.1109/18.796380.  Google Scholar [4] F. Fiedler, J. Jedwab and M. G. Parker, A framework for the construction of Golay sequences,, IEEE Trans. Inform. Theory, 54 (2008), 3113.  doi: 10.1109/TIT.2008.924667.  Google Scholar [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.  doi: 10.1016/j.jcta.2007.10.001.  Google Scholar [6] M. J. E. Golay, Complementary series,, IEEE Trans. Inform. Theory, 7 (1961), 82.   Google Scholar [7] S. Litsyn, Peak Power Control in Multi-carrier Communications,, Cambridge Univ. Press, (2007).   Google Scholar [8] M. G. Parker and C. Riera, Generalised complementary arrays,, IMA Int. Conf. Crypt. Coding, (2011), 41.  doi: 10.1007/978-3-642-25516-8_4.  Google Scholar [9] M. G. Parker and C. Tellambura, Golay-Davis-Jedwab complementary sequences and Rudin-Shapiro constructions,, in Int. Symp. Inform. Theory, (2001).   Google Scholar [10] M. G. Parker and C. Tellambura, A construction for binary sequence sets with low peak-to-average power ratio,, Reports in Informatics, (2003).   Google Scholar [11] K. G. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation,, IEEE Trans. Inform. Theory, 46 (2000), 104.  doi: 10.1109/18.817512.  Google Scholar [12] K.-U. Schmidt, On cosets of the generalized first-order Reed-Muller code with low PMEPR,, IEEE Trans. Inform. Theory, 52 (2006), 3220.  doi: 10.1109/TIT.2006.876252.  Google Scholar [13] K.-U. Schmidt, Complementary sets, generalized Reed-Muller codes, and power control for OFDM,, IEEE Trans. Inform. Theory, 53 (2007), 808.  doi: 10.1109/TIT.2006.889723.  Google Scholar [14] T. E. Stinchcombe, Aperiodic autocorrelations of length $2^m$ sequences, complementarity, and power control for OFDM,, Ph.D. thesis, (2000).   Google Scholar [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.  doi: 10.1109/TIT.2014.2300867.  Google Scholar [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.   Google Scholar [17] G. Wu and M. G. Parker, A complementary construction using mutually unbiased bases,, Crypt. Commun., 6 (2014), 3.  doi: 10.1007/s12095-013-0095-9.  Google Scholar [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.   Google Scholar [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.  doi: 10.1109/TIT.2010.2090230.  Google Scholar
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