If you're a PlayStation Plus subscriber in North America, you might want to head straight to the PlayStation Store, as the network stress test version of Dark Souls III is now available for download.
The download can be found here, and it weighs a mere 2.9 Gb.
After downloading, you won't be able to play immediately, as the servers are down, and will go up according to the following schedule:
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Test #1
- Friday, October 16th, 2015 at 8:00 p.m. to 11:00 p.m. PDT
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Test #2
- Saturday, October 17th, 2015 at 8:00 p.m. to 11:00 p.m. PDT
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Test #3
- Sunday, October 18th, 2015 at 8:00 p.m. to 11:00 p.m. PDT
Interestingly, the download page also mentions that the game supports between two and six players online.
Update: unfortunately Sony dropped the axe on the download, indicating that it probably wasn't supposed to be released just yet. That said, it's still available in the library of those who already "purchased" it.