Wikipedia writes that
"if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by $k$ arbitrary functions and their derivatives up to order $m$, then the functional derivatives of $L$ satisfy a system of $k$ differential equations."
Such a transformation may look like $$\delta\varphi = \text{i} \omega_a(x)T^a F[\varphi] + \text{i}\sum\limits_{j=1}^{m} F^{\mu_1..\mu_j}[\varphi](\partial_{\mu_1}..\partial_{\mu_j})\omega_a(x)T^a \quad a=1,..,k$$
They specify the parameters $k$ and $m$, however only state how $k$ is interpreted. What about $m$? How does the choice of $m$ affect Noether's second theorem?