That circuit doesn't give turns ratio or a rectifier and filter...
It has no current sense, so it's very likely to explode at... anything. It has no regulation, it runs fully open loop. The only reason it even starts up in the first place is because of soft start, which is very wishful thinking.
The gate driver circuit is also inconsistent, but it'll probably go slow enough you won't notice.
Given the voltages in the text, I would guess it needs a full wave rectifier, 1:4 turns ratio. A typical FBT core has probably a 13mm peg, or 132 mm^2 cross section, which will handle 0.2-0.3T being ferrite. With a maximum 16VDC input, you need 2.4 turns minimum per half of the primary winding. 3 or 4 turns would be fine.
So your winding should be: 4+4 turns primary, 16+16 turns secondary.
For the primary, wind one flat layer of many wires in parallel, or copper strip (wrapped with tape for insulation). It is better to use many thinner wires in parallel, than one single solid piece rated for the current.
For the secondary, use fewer pieces of wire, and wind the 16+16 turns directly on top of the primary as closely as possible.
With center taps, the primary and secondary are themselves composed of halves. These should be interleaved as layers, too.
So you should have:
4 turns primary half
16 turns secondary half
4 turns primary other half
16 turns secondary other half
and connect the halves in series, finish to start, so the wire is always going around the same direction.
The switching layout has to be extremely dense and low inductance, otherwise you'll have transistors blowing up themselves, not sharing current evenly, oscillating, etc.
You should have an oscilloscope to measure these waveforms as well.
Tim