Numerical simulations including magnetic fields have become important in many fields of astrophysics. Evolution of magnetic fields by the constrained-transport algorithm preserves magnetic divergence to machine precision and thus represents one preferred method for the inclusion of magnetic fields in simulations. We show that constrained transport can be implemented with volume-centered fields and hyperresistivity on a high-order finite-difference stencil. In addition, the finite-difference coefficients can be tuned to enhance high-wavenumber resolution. Similar techniques can be used for the interpolations required for dealiasing corrections at high wavenumber. Together, these measures yield an algorithm with a wavenumber resolution that approaches the theoretical maximum achieved by spectral algorithms. Because this algorithm uses finite differences instead of fast Fourier transforms, it runs faster and is not restricted to periodic boundary conditions. In addition, since the finite differences are spatially local, this algorithm is easily scalable to thousands of processors. We demonstrate that, for low-Mach number turbulence, the results agree well with a high-order, non-constrained-transport scheme with Poisson divergence cleaning.