A k-ary deBruijn sequence of order n is a sequence of period k n which contains each k-ary n-tuple exactly once during each period. DeBruijn sequences are named after the Dutch mathematician Nicholas deBruijn. In 1946 he discovered a formula giving the number of k-ary deBruijn sequences of order n, and proved that it is given by \(((k-1)!) ^{k^{n-1}}\cdot{k^{k^{n-1}-n}}\). The result was, however, first obtained more than 50 years earlier, in 1894, by the French mathematician C. Flye-Sainte Marie.
For most applications binary deBruijn sequences are the most important. The number of binary deBruijn sequences of period \(2^n\) is \(2^{2^{n-1}-n}\). An example of a binary deBruijn sequence of period \(2^4=16\) is \(\{s_t\}=0000111101100101\). All binary 4-tuples occur exactly once during a period of the sequence. In general, binary deBruijn sequences are balanced, containing the same number of 0's and 1's in a period, and they satisfy many randomness criteria, although they may be...
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References
Fredricksen, H. (1982). “A survey of full length nonlinear shift register cycle algorithms.” SIAM Review, 24 (2), 195–221.
Golomb, S.W. (1982). Shift Register Sequences. Aegean Park Press, Laguna Hills, CA.
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Helleseth, T. (2005). DeBruijn sequence. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_98
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