Abstract: We present mosaic theorem for binary sequence and multiple sampling sequence. Main conclusion includes: multiple sampling sequence has nothing to do with its original state, and g(x^<sup>N₁) is one of itsgenerator polynomial, and there is a SSCO that its minimum polynomial m_c(x)=g(x^<sup>N₁); When the number of “1” is power of 2 in the controlled sequence, the minimum polynomial of multiple sampling sequence is g^t(x), its period is 2^r(2ⁿ-1); Specially, When the controlled sequence is m-Sequence and (m,n)=1,m≤n/2, the minimum polynomial of multiple sampling sequence is m_c(x)=g^t(x), its period is 2^m-1(2ⁿ-1),2^m-2＜t≤2^m-1.

Key words: mosaic theorem, period, minimum polynomial, trace