I have a 2.4 GHz transceiver that outputs 4.5 DBm. I wish to amplify the signal using a 15.5 DBi antenna so that the signal reaches 20 DB, the maximum legal limit. Preferably, the antenna would be of a type that I can make myself. Any suggestions on what type and design of antenna that fits my needs?
An antenna can't "amplify" a signal, it can only concentrate it in certain directions. If you want a peak gain of 15 dB (about 30x the power density of an isotropic radiator), you can only get it over a very small portion of the sphere (about 3% or less).
When we talk about "omnidirectional" antennas, usually we mean that the antenna element is vertical, and the antenna radiates equally in all directions in the horizontal plane. But the gain has to come from somewhere, and in this case it comes from compressing the pattern vertically, so that the signal strength decreases if you go above or below the plane of the antenna. At 15dB, the pattern is very squished, and the vertical beamwidth is quite small (around 6°).
That said, if you want one, there is a "simple" design that will at least come close: it's a collinear array of around 25 elements (roughly 1.5 meters tall, for 2.4 GHz).