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International Journal of Combinatorics
Volume 2010, Article ID 851857, 21 pages
http://dx.doi.org/10.1155/2010/851857
Research Article

Classification of Base Sequences 𝐵 𝑆 ( 𝑛 + 1 , 𝑛 )

Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

Received 6 February 2010; Revised 8 April 2010; Accepted 12 April 2010

Academic Editor: Masaaki Harada

Copyright © 2010 Dragomir Ž. Ðoković. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Base sequences BS ( 𝑛 + 1 , 𝑛 ) are quadruples of { ± 1 } -sequences ( 𝐴 ; 𝐵 ; 𝐶 ; 𝐷 ) , with A and B of length 𝑛 + 1 and C and D of length n, such that the sum of their nonperiodic autocor-relation functions is a 𝛿 -function. The base sequence conjecture, asserting that BS ( 𝑛 + 1 , 𝑛 ) exist for all n, is stronger than the famous Hadamard matrix conjecture. We introduce a new definition of equivalence for base sequences BS ( 𝑛 + 1 , 𝑛 ) and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of BS ( 𝑛 + 1 , 𝑛 ) for 𝑛 3 0 . As the number of equivalence classes grows rapidly (but not monotonically) with n, the tables in the paper cover only the cases 𝑛 1 3 .