November  2013, 7(4): 379-407. doi: 10.3934/amc.2013.7.379

Small Golay sequences

1. 

Wesley College, 120 N State St, Dover, DE 19901, United States

Received  December 2010 Revised  July 2013 Published  October 2013

We enumerate $H$-phase Golay sequences for $H\le 36$ and lengths up to 33. Our enumeration method is based on filtering by the power spectra. Some of the hexaphase Golay sequence pairs are new. We provide a compact way to reconstruct all these Golay sequences from specific Golay arrays. The Golay arrays are part of the three-stage construction introduced by Fiedler, Jedwab, and Parker. All such minimal Golay arrays can be constructed from a small set of Golay sequence pairs with binary, quaternary, or hexaphase alphabet adjoining 0. We also prove some non-existence results for Golay sequences when $H/2$ is odd.
Citation: Frank Fiedler. Small Golay sequences. Advances in Mathematics of Communications, 2013, 7 (4) : 379-407. doi: 10.3934/amc.2013.7.379
References:
[1]

P. B. Borwein and R. A. Ferguson, A complete description of Golay pairs for lengths up to 100, Math. Comp., 73 (2004), 967-985. doi: 10.1090/S0025-5718-03-01576-X.

[2]

R. Craigen, W. Holzmann and H. Kharaghani, Complex Golay sequences: structure and applications, Discrete Math., 252 (2002), 73-89. doi: 10.1016/S0012-365X(01)00162-5.

[3]

J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes, IEEE Trans. Inform. Theory, 45 (1999), 2397-2417. doi: 10.1109/18.796380.

[4]

S. Eliahou, M. Kervaire and B. Saffari, A new restriction on the length of Golay complementary sequences, J. Comb. Theory Ser. A, 55 (1990), 49-59. doi: 10.1016/0097-3165(90)90046-Y.

[5]

S. Eliahou, M. Kervaire and B. Saffari, On Golay polynomial pairs, Adv. Appl. Math., 12 (1991), 235-292. doi: 10.1016/0196-8858(91)90014-A.

[6]

F. Fiedler and J. Jedwab, How do more Golay sequences arise? IEEE Trans. Inform. Theory, 52 (2006), 4261-4266. doi: 10.1109/TIT.2006.880024.

[7]

F. Fiedler, J. Jedwab and M. G. Parker, A multi-dimensional approach to the construction and enumeration of Golay complementary sequences, J. Comb. Theory Ser. A, 115 (2008), 753-776. doi: 10.1016/j.jcta.2007.10.001.

[8]

F. Fiedler, J. Jedwab and A. Wiebe, A new source of seed pairs for Golay sequences of length $2^m$, J. Comb. Theory Ser. A, 117 (2010), 589-597. doi: 10.1016/j.jcta.2009.12.009.

[9]

R. G. Gibson and J. Jedwab, Quaternary Golay sequence pairs I: even length, Des. Codes Cryptogr., 59 (2011), 131-146. doi: 10.1007/s10623-010-9471-z.

[10]

R. G. Gibson and J. Jedwab, Quaternary Golay sequence pairs II: odd length, Des. Codes Cryptogr., 59 (2011), 147-157. doi: 10.1007/s10623-010-9472-y.

[11]

M. J. E. Golay, Complementary series, IRE Trans. Inform. Theory, IT-7 (1961), 82-87.

[12]

W. H. Holzmann and H. Kharaghani, A computer search for complex Golay sequences, Australasian J. Comb., 10 (1994), 251-258.

[13]

J. Jedwab and M. G. Parker, There are no Barker arrays having more than two dimensions, Des. Codes Cryptogr., 43 (2007), 79-84. doi: 10.1007/s10623-007-9060-y.

[14]

J. Jedwab and M. G. Parker, Golay complementary array pairs, Des. Codes Cryptogr., 44 (2007), 209-216. doi: 10.1007/s10623-007-9088-z.

[15]

J. Jedwab and M. G. Parker, Binary length 10 Golay sequences are equivalent, personal communication, 2009.

[16]

T. Y. Lam and K. H. Leung, On vanishing sums of roots of unity, J. Algebra, 224 (2000), 91-109. doi: 10.1006/jabr.1999.8089.

show all references

References:
[1]

P. B. Borwein and R. A. Ferguson, A complete description of Golay pairs for lengths up to 100, Math. Comp., 73 (2004), 967-985. doi: 10.1090/S0025-5718-03-01576-X.

[2]

R. Craigen, W. Holzmann and H. Kharaghani, Complex Golay sequences: structure and applications, Discrete Math., 252 (2002), 73-89. doi: 10.1016/S0012-365X(01)00162-5.

[3]

J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes, IEEE Trans. Inform. Theory, 45 (1999), 2397-2417. doi: 10.1109/18.796380.

[4]

S. Eliahou, M. Kervaire and B. Saffari, A new restriction on the length of Golay complementary sequences, J. Comb. Theory Ser. A, 55 (1990), 49-59. doi: 10.1016/0097-3165(90)90046-Y.

[5]

S. Eliahou, M. Kervaire and B. Saffari, On Golay polynomial pairs, Adv. Appl. Math., 12 (1991), 235-292. doi: 10.1016/0196-8858(91)90014-A.

[6]

F. Fiedler and J. Jedwab, How do more Golay sequences arise? IEEE Trans. Inform. Theory, 52 (2006), 4261-4266. doi: 10.1109/TIT.2006.880024.

[7]

F. Fiedler, J. Jedwab and M. G. Parker, A multi-dimensional approach to the construction and enumeration of Golay complementary sequences, J. Comb. Theory Ser. A, 115 (2008), 753-776. doi: 10.1016/j.jcta.2007.10.001.

[8]

F. Fiedler, J. Jedwab and A. Wiebe, A new source of seed pairs for Golay sequences of length $2^m$, J. Comb. Theory Ser. A, 117 (2010), 589-597. doi: 10.1016/j.jcta.2009.12.009.

[9]

R. G. Gibson and J. Jedwab, Quaternary Golay sequence pairs I: even length, Des. Codes Cryptogr., 59 (2011), 131-146. doi: 10.1007/s10623-010-9471-z.

[10]

R. G. Gibson and J. Jedwab, Quaternary Golay sequence pairs II: odd length, Des. Codes Cryptogr., 59 (2011), 147-157. doi: 10.1007/s10623-010-9472-y.

[11]

M. J. E. Golay, Complementary series, IRE Trans. Inform. Theory, IT-7 (1961), 82-87.

[12]

W. H. Holzmann and H. Kharaghani, A computer search for complex Golay sequences, Australasian J. Comb., 10 (1994), 251-258.

[13]

J. Jedwab and M. G. Parker, There are no Barker arrays having more than two dimensions, Des. Codes Cryptogr., 43 (2007), 79-84. doi: 10.1007/s10623-007-9060-y.

[14]

J. Jedwab and M. G. Parker, Golay complementary array pairs, Des. Codes Cryptogr., 44 (2007), 209-216. doi: 10.1007/s10623-007-9088-z.

[15]

J. Jedwab and M. G. Parker, Binary length 10 Golay sequences are equivalent, personal communication, 2009.

[16]

T. Y. Lam and K. H. Leung, On vanishing sums of roots of unity, J. Algebra, 224 (2000), 91-109. doi: 10.1006/jabr.1999.8089.

[1]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015

[2]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure and Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271

[3]

Pinhui Ke, Yueqin Jiang, Zhixiong Chen. On the linear complexities of two classes of quaternary sequences of even length with optimal autocorrelation. Advances in Mathematics of Communications, 2018, 12 (3) : 525-539. doi: 10.3934/amc.2018031

[4]

Oǧuz Yayla. Nearly perfect sequences with arbitrary out-of-phase autocorrelation. Advances in Mathematics of Communications, 2016, 10 (2) : 401-411. doi: 10.3934/amc.2016014

[5]

Samuel T. Blake, Thomas E. Hall, Andrew Z. Tirkel. Arrays over roots of unity with perfect autocorrelation and good ZCZ cross-correlation. Advances in Mathematics of Communications, 2013, 7 (3) : 231-242. doi: 10.3934/amc.2013.7.231

[6]

Lin Yi, Xiangyong Zeng, Zhimin Sun, Shasha Zhang. On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021019

[7]

Pinhui Ke, Panpan Qiao, Yang Yang. On the equivalence of several classes of quaternary sequences with optimal autocorrelation and length $ 2p$. Advances in Mathematics of Communications, 2022, 16 (2) : 285-302. doi: 10.3934/amc.2020112

[8]

Simone Fiori. Auto-regressive moving-average discrete-time dynamical systems and autocorrelation functions on real-valued Riemannian matrix manifolds. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2785-2808. doi: 10.3934/dcdsb.2014.19.2785

[9]

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

[10]

Mohammed Al-Azba, Zhaohui Cen, Yves Remond, Said Ahzi. Air-Conditioner Group Power Control Optimization for PV integrated Micro-grid Peak-shaving. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3165-3181. doi: 10.3934/jimo.2020112

[11]

Hongming Yang, C. Y. Chung, Xiaojiao Tong, Pingping Bing. Research on dynamic equilibrium of power market with complex network constraints based on nonlinear complementarity function. Journal of Industrial and Management Optimization, 2008, 4 (3) : 617-630. doi: 10.3934/jimo.2008.4.617

[12]

Ernesto Aranda, Pablo Pedregal. Constrained envelope for a general class of design problems. Conference Publications, 2003, 2003 (Special) : 30-41. doi: 10.3934/proc.2003.2003.30

[13]

Bingsheng Shen, Yang Yang, Ruibin Ren. Three constructions of Golay complementary array sets. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022019

[14]

Konstantinos Drakakis, Francesco Iorio, Scott Rickard, John Walsh. Results of the enumeration of Costas arrays of order 29. Advances in Mathematics of Communications, 2011, 5 (3) : 547-553. doi: 10.3934/amc.2011.5.547

[15]

Konstantinos Drakakis, Francesco Iorio, Scott Rickard. The enumeration of Costas arrays of order 28 and its consequences. Advances in Mathematics of Communications, 2011, 5 (1) : 69-86. doi: 10.3934/amc.2011.5.69

[16]

Johan Chrisnata, Han Mao Kiah, Sankeerth Rao Karingula, Alexander Vardy, Eitan Yaakobi Yao, Hanwen Yao. On the number of distinct k-decks: Enumeration and bounds. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021032

[17]

Linda J. S. Allen, Vrushali A. Bokil. Stochastic models for competing species with a shared pathogen. Mathematical Biosciences & Engineering, 2012, 9 (3) : 461-485. doi: 10.3934/mbe.2012.9.461

[18]

Thomas Demoor, Dieter Fiems, Joris Walraevens, Herwig Bruneel. Partially shared buffers with full or mixed priority. Journal of Industrial and Management Optimization, 2011, 7 (3) : 735-751. doi: 10.3934/jimo.2011.7.735

[19]

Min Ji, Xinna Ye, Fangyao Qian, T.C.E. Cheng, Yiwei Jiang. Parallel-machine scheduling in shared manufacturing. Journal of Industrial and Management Optimization, 2022, 18 (1) : 681-691. doi: 10.3934/jimo.2020174

[20]

Feishe Chen, Lixin Shen, Yuesheng Xu, Xueying Zeng. The Moreau envelope approach for the L1/TV image denoising model. Inverse Problems and Imaging, 2014, 8 (1) : 53-77. doi: 10.3934/ipi.2014.8.53

2020 Impact Factor: 0.935

Metrics

  • PDF downloads (120)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]