  In this paper, we make some progress towards a well-known conjecture on the
minimum weights of binary cyclic codes with two primitive nonzeros. We also
determine the Walsh spectrum of $\Tr(x^d)$ over $\F_{2^{m}}$ in the case where
$m=2t$, $d=3+2^{t+1}$ and $\gcd(d, 2^{m}-1)=1$.
