Antennas operating at the high-frequency (HF) band (3-30 MHz) are frequently electrically small due to the large wavelength of electromagnetic waves (10-100 m). However, the bandwidth-efficiency products of passively matched electrically small antennas (ESAs) are fundamentally limited. Wideband HF waveforms using bandwidths of 24 kHz or more have recently received significant attention in military communications applications. Efficiently radiating such signals from conventional passive ESAs is very challenging due to fundamental physical limits on bandwidth-efficiency products of ESAs. However, active antennas are not subject to the same constraints. In this work, we present the design and experimental characterization of a high-power, active ESA with enhanced bandwidth-efficiency product compared to {that of} passively matched ESAs. Specifically, the proposed active ESA can radiate wideband HF signals with banwidths of 24 kHz or more, with total efficiencies up to 80$\%$, and radiated power levels approaching 100 W. Our approach uses a highly-efficient, integrated class-E switching circuit specifically designed to drive an electrically small, high-Q HF antenna over a bandwidth exceeding 24 kHz. Using a high-Q RLC antenna model, we have successfully demonstrated wideband binary ASK, PSK, and FSK modulations with the proposed class-E switching architecture. Experimental results indicate that the bandwidth-efficiency product of this class-E active antenna is 5.4-9.8 dB higher than that of an equivalent passive design with the same data rate, and bit-error-rate (BER).