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Transpositions matrix T r = i = 1 , n j = 1 , n {\displaystyle Tr=_{\begin{smallmatrix}i={1,n}\\j={1,n}\end{smallmatrix}}} is square n × n {\displaystyle n\times n} matrix, n = 2 m {\displaystyle n=2^{m}} , m ∈ N {\displaystyle m\in N} , which elements are obtained from the elements of given n-dimensional vector X = i = 1 , n {\displaystyle X=_{\begin{smallmatrix}i={1,n}\end{smallmatrix}}} as follows: T r i , j = x ⊕ + 1 {\displaystyle Tr_{i,j}=x_{\oplus +1}} , where ⊕ {\displaystyle \oplus } denotes operation "bitwise Exclusive or". The rows and columns of transpositions matrix consists permutation of elements of vector X {\displaystyle X} , as there are n/2 transpositions between every two rows or columns of the matrix