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The New Zealand number one leads by a shot after opening with a five-under 67.
The top three players not already qualified for the British Open, who finish in the top 10, will earn a place at Carnoustie later this month.
"I'd love to be in the Open and Carnoustie is one of the best golf courses in the world I think and I'd love the chance to play there in an Open," Fox said after his first round.
"As good a start as I can have, really, and hopefully three more rounds and if I can sneak in that top five like last year and get an Open spot, that would be great."
"I drove it nice. I think I probably only hit five or six fairways but missed them in the right places and hit a lot of good drives that just ran through fairways and gave myself a lot of chances," he said.