  Consider a random $k$-CNF formula $F_{k}(n, rn)$ with $n$ variables and $rn$
clauses. For every truth assignment $\sigma\in \{0, 1\}^{n}$ and every clause
$c=\ell_{1}\vee\cdots\vee\ell_{k}$, let $d=d(\sigma, c)$ be the number of
satisfied literal occurrences in $c$ under $\sigma$. For fixed $\beta>-1$ and
$\lambda>0$, we take $\omega(\sigma, c)=0$, if $d=0$; $\omega(\sigma,
c)=\lambda(1+\beta)$, if $d=1$ and $\omega(\sigma, c)=\lambda^{d}$, if $d>1$.
  Applying the above weighting scheme, we get that if $F_{k}(n, rn)$ is
unsatisfiable with probability tending to one as $n\rightarrow\infty$, then
$r\geq2.83, 8.09, 18.91, 40.81, 84.87$ for $k=3, 4, 5, 6$ and $7,$
respectively.
