About the ordinances of the vectors of the n-dimensional Boolean cube in accordance with their weights
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The problem "Given a Boolean function f of n variables by its truth table vector. Find (if exists) a vector α∈{0,1}^n of maximal (or minimal) weight, such that f(α)= 1." arises in computing the algebraic degree of Boolean functions or vectorial Boolean functions called S-boxes. The solutions to this problem have useful generalizations and applications. To find effective solutions we examine the ways of ordering the vectors of the Boolean cube in accordance with their weights. The notion "k-th layer" of the n-dimensional Boolean cube is involved in the definition and examination of the "weight order" relation. It is compared with the known relation "precedes". We enumerate the maximum chains for both relations. An algorithm that generates the vectors of the n-dimensional Boolean cube in accordance with their weights is developed. The lexicographic order is chosen as a second criterion for an ordinance of the vectors of equal weights. The algorithm arranges the vectors in a unique way called a weight-lexicographic order. It is represented by the serial numbers of the vectors, instead of the vectors itself. Its time and space complexities are Θ (2^n), i.e., of linear type with respect to the size of the output. The obtained results are summarized and added as a new sequence (A294648) in the OEIS.
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