Multiplexed superconducting quantum interference device (SQUID) readout systems are critical for measuring large arrays of superconducting transition-edge sensors (TES). We demonstrate a code-division SQUID multiplexing (CDM) architecture that is modulated by Walsh codes. Measurements and simulations of a prototype multiplexer show that this modulation scheme is not degraded by SQUID-noise aliasing, suppresses parasitic pickup, and has low levels of crosstalk. These properties enable this architecture to scale to large TES arrays. Furthermore, CDM modulation suppresses the 1/f knee in the noise to below 20 mHz, suggesting the use of this circuit for low-frequency-noise mitigation in more general SQUID applications.