De Bruijn Sequence

Related Concepts

Boolean Functions; Linear Complexity; Sequences; Maximal-Length Linear Sequences; Nonlinear Feedback Function

Definition

A k-ary de Bruijn sequence of order n is a sequence of period k ⁿ, which contains each k-ary n-tuple exactly once during each period.

Theory and Application

De Bruijn sequences are named after the Dutch mathematician Nicholas de Bruijn. In 1946, he discovered a formula giving the number of k-ary de Bruijn 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 de Bruijn sequences are the most important. The number of binary de Bruijn sequences of period 2ⁿ is ${2}^{{2}^{n-1}-n }$. An example of a binary de Bruijn sequence of period ${2}^{4}\,=\,16$ is $\{{s}_{t}\}\,=\,0000111101100101$. All binary 4-tuples occur exactly once during a...