Multivariate mutual information provides a conceptual framework for characterizing higher-order interactions in complex systems. Two well-known measures of multivariate information---total correlation and dual total correlation---admit a spectrum of measures with varying sensitivity to intermediate orders of dependence. Unfortunately, these intermediate measures have not received much attention due to their opaque representation of information. Here we draw on results from matroid theory to show that these measures are closely related to error-correcting codes. This connection allows us to derive the class of global maximizers for each measure, which coincide with maximum distance separable codes of order $k$. In addition to deepening the understanding of these measures and multivariate information more generally, we use these results to show that previously proposed bounds on information geometric quantities are tight at the extremes.