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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

   

 

A complete description of Golay pairs for lengths up to 100


Authors: P. B. Borwein and R. A. Ferguson
Journal: Math. Comp. 73 (2004), 967-985
MSC (2000): Primary 11B83, 05B20; Secondary 94A11, 68R05
Published electronically: July 1, 2003
MathSciNet review: 2031419
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Abstract: In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length $n$ with complementary autocorrelation is at worst a $2^{\frac{3n}{2}-6}$ problem. Andres, in his 1977 master's thesis, developed an algorithm which reduced this to a $2^{\frac{n}{2}-1}$ search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a $2^{\frac{n}{2}-5}$ search at length 82. We find no new pairs at the outstanding lengths 74 and 82. In extending the theory of composition, we are able to obtain a closed formula for the number of pairs of length $2^kn$ generated by a primitive pair of length $n$. Combining this with the results of searches at all allowable lengths up to 100, we identify five primitive pairs. All others pairs of lengths less than 100 may be derived using the methods outlined.


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Additional Information

P. B. Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia V5A 1S6 Canada
Email: pborwein@cecm.sfu.ca

R. A. Ferguson
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia V5A 1S6 Canada
Email: rferguson@pims.math.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01576-X
PII: S 0025-5718(03)01576-X
Keywords: Complementary pairs, composition of sequences
Received by editor(s): December 10, 2001
Received by editor(s) in revised form: November 28, 2002
Published electronically: July 1, 2003
Additional Notes: Research of the authors was supported in part by grants from NSERC of Canada and MITACS Symbolic Analysis Project
Article copyright: © Copyright 2003 Copyright retained by the authors