Visible Light Communication LED Turnoff Time Problem
White LEDs are the wrong choice for communication.  This is because they are a two-part system.  Inside is a blue or near UV LED and a mix of phosphors.  The LED excites the phosphors, which then emits various colors of light.  The result is balanced to look roughly white.

The reason for the lag is that the phosphors will continue to re-emit the energy for a short time after the LED is turned off.

Use anything that is a bare LED.  This also has the advantage of all the light being within a very narrow spectral band.  That makes it possible to filter out much of the ambient light at the receiver.

IR LEDs are usually used for this since the ambient level at near IR tends to be lower than the visible level, IR LEDs are efficient, and materials are available that block most visible light but let the IR pass.  Wavelengths are usually in the 900-1000 nm range.