  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 $f(\alpha)= 1$." is considered here. It is closely
related to the problem of fast computing the algebraic degree of Boolean
functions. It is an important cryptographic parameter used in the design of
S-boxes in modern block ciphers, PRNGs in stream ciphers, at Reed-Muller codes,
etc. To find effective solutions to this problem we explore the orders of the
vectors of the $n$-dimensional Boolean cube $\{0,1\}^n$ in accordance with
their weights. The notion of "$k$-th layer" of $\{0,1\}^n$ is involved in the
definition and examination of the "weight order" relation. It is compared with
the known relation "precedes". Several enumeration problems concerning these
relations are solved and the corresponding comments were added to 3 sequences
in the On-line Encyclopedia of Integer Sequences (OEIS). One special order
(among the numerous weight orders) is defined and examined in detail. The
lexicographic order is a second criterion for an ordinance of the vectors of
equal weights. So a total order called Weight-Lexicographic Order (WLO) is
obtained. Two algorithms for generating the WLO sequence and two algorithms for
generating the characteristic vectors of the layers are proposed. Their results
were used in creating 2 new sequences: A294648 and A305860 in the OEIS. Two
algorithms for solving the problem considered are developed--the first one
works in a byte-wise manner and uses the WLO sequence, and the second one works
in a bitwise manner and uses the characteristic vector as masks. The
experimental results after many tests confirm the efficiency of these
algorithms. Some other applications of the obtained algorithms are also
discussed--for example, when representing, generating and ranking other
combinatorial objects.
