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.  doi: 10.1090/S0025-5718-03-01576-X.  Google Scholar

[2]

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

[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.  doi: 10.1109/18.796380.  Google Scholar

[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.  doi: 10.1016/0097-3165(90)90046-Y.  Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1016/j.jcta.2007.10.001.  Google Scholar

[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.  doi: 10.1016/j.jcta.2009.12.009.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

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.  doi: 10.1090/S0025-5718-03-01576-X.  Google Scholar

[2]

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

[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.  doi: 10.1109/18.796380.  Google Scholar

[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.  doi: 10.1016/0097-3165(90)90046-Y.  Google Scholar

[5]

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

[6]

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

[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.  doi: 10.1016/j.jcta.2007.10.001.  Google Scholar

[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.  doi: 10.1016/j.jcta.2009.12.009.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

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