How to drive 128 LEDs
The most efficient way to drive such a big number of LEDs will probably be to use a constant current generator (there are many ICs for that, many dedicated to LED applications, as for example the AP8800, which is a 350 mA converter for LEDs — this is a fairly standard current value for LED lighting applications, what current do you actually need for your UV LEDs?), and then connect the LEDs in series as strips of (for example) 8, each strip having its own constant current generator. Of course you will need to power the board with a relatively high voltage (e.g., 24V), but this should generally not be a problem.

A very simple solution for your 20 mA, 3.8V LEDs could be a linear constant current regulator like the CL520 (a small transistor-like device, which only takes an external 100 nF capacitor to work), or a constant current diode, like the NSI45020T1G: if you take a standard 48V power supply, one CL520 could handle 12 LEDs in series (the NSI45020T1G is limited to 45V). If the LEDs have the highest VF, with 48V input and 12 LEDs (total VF = 45.6 V) the regulator has to dissipate 48 mW, which would only lead to temperature increase of around 6 °C with the small TO-92 package. If the LEDs have the lowest VF (total VF = 38.4 V), the total dissipation would be 192 mW, with a temperature increase around 25 °C (which is still acceptable). For LEDs with higher currents, however, such devices that just dissipate the excess voltage as heat would heat up way too much, that's why devices like the AP8800 are implemented as switching step-down converters (using a coil).