A novel weighting scheme for random $k$-SAT

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 $c={\mathrm{\ell }}_1\vee \cdots \vee {\mathrm{\ell }}_k$, 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 $n\to \mathrm{\infty }$, then $r\ge 2.83,8.09,18.91,40.81,84.87$ for $k=3,4,5,6$ and $7,$ respectively.