Бакоев,
Валентин
(2018)
About the ordinances of the vectors of the ndimensional Boolean cube in accordance with their weights
Cornell University Library, https://arxiv.org/abs/1811.04421
The problem ''Given a Boolean function f of n variables by its truth table vector. Find (if exists) a vector $\alpha \in \{0,1\}^n$ of maximal (or minimal) weight, such that $f(\alpha)= 1$.'' arises in computing the algebraic degree of Boolean functions or vectorial Boolean functions called Sboxes. 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 ''kth layer'' of the ndimensional Boolean cube is involved in the definition and examination of the ''weight order'' relation. It is compared with the known relation ''precedes''. The maximum chains for both relations are enumerated. An algorithm that generates the vectors of the ndimensional 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 weightlexicographic order. It is represented by the serial numbers of the vectors, instead of the vectors itself. Its time and space complexities are $\Theta (2^n)$, 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.
Статия
Boolean cube, binary vector, serial number, lexicographic order, weight order, maximum chains enumerating, weightlexicographic order generating, power set generating, ranking
Природни науки, математика и информатика
Природни науки, математика и информатика
Информатика и компютърни науки
Natural sciences, mathematics and informatics
Natural sciences, mathematics and informatics
Informatics and Computer Science
Издадено
22586
Валентин Бакоев
