  In this paper, we construct the chain level representation in the lambda
algebra of the Lannes-Zarati homomorphism for $\mathbb{F}_2$ as well as for
$\tilde{H}^*B\mathbb{Z}/2$. Using this construction, we show that the sixth
Lannes-Zarati homomorphism for $\mathbb{F}_2$ is trivial at all stems $t$ for
$t\leq 114$. Furthermore, using this method, we also investigate the behavior
of the Lannes-Zarati homomorphis for $\tilde{H}^*B\mathbb{Z}/2$ at the
homological $s\leq 4$.
