We investigate the cofinality of the partial order $\mathscr{N}^\kappa$ of functions from a regular cardinal $\kappa$ into the ideal $\mathscr{N}$ of Lebesgue measure zero subsets of $\mathbf{R}$. We show that when add$(\mathscr{N}) = \kappa$ and the covering lemma holds with respect to an inner model of GCH, then $\mathrm{cf}(\mathscr{N}^\kappa) = \max \{\mathrm{cf}(\kappa^\kappa), \mathrm{cf}(\lbrack \mathrm{cf}(\mathscr{N})\rbrack^\kappa)\}$. We also give an example to show that the covering assumption cannot be removed.