I am remote ballasting a 175 watt mercury vapor lamp off of an M57 ballast and I was wondering, to get the maximum brightness from the lamp, what the maximum run between the ballast and the bulb could be using 16 guage wire. Currently the run is approximately 16 feet.

Thanks.

Mercury ballast (H...) should be able to tolerate at least extra 10V drop, MH ballast even way more (above 20V; due to higher OCV). But you should be aware, then extra drop cause extra power dissipation, so the system power input would increase. Practically i would not go above 10V wiring drop, what mean wire resistance up to 6 Ohm for 175W MV/MH should be OK. I don't know the cross section of the AWG16, but for here (EU) commonly used 1.5mm^2 mains installation cables it mean about 225m, so about 750 feet cable length...
For thinner, 1mm^2 it would be accordingly shorter, so 150m/500feet...

That answers it, thanks. Another question....

If I wanted to use a 175 watt probe start MH lamp instead off the same ballast, would I need a 4Kv socket?

No, probe start does not have a starting pulse, so as long as the cable is rated for a few hundred volts (most if not all lamp cords or SJT extension cords are >300V)

For gauging conversions and resistance per foot, there's a table at Wikipedia.

And if you would like to remotely ballast pulse-start lamps, it is good idea to use ballast without (or with deactivated) ignitor and place the superimposed type ignitor close to the lamp (into the fixture), so the long wiring would not be exposed to high voltage ignition pulses.
Another option is to keep the ignitor in the ballast box, but then you have to check the wiring capacitance, as each ignitor (or ballast + ignitor combination) have specified maximum load capacitance (for standard ignitors <200pF), so for longer wiring you should use "long range" ignitor types (available with allowed load from 2 to more then 10nF). You should count ~100pF/1meter (30pF/feet), so for standard ignitor (<200pF) the length should not exceed 2m (6 feet), for typical semiparallel long range (2nF load rated) 20m (60 feet).
The resistance limit is then similar as for MV/probe MH (voltage drop should be below 10% of arc voltage, what mean ~6 Ohm for 150W MH), regardless of ignitor connection.

Great information, thanks.