が早いか means "no sooner had… than": the second thing happened the instant the first one did, almost at the same time. It goes after a verb in the dictionary form, less often the た-form, and it is literary – you meet it in novels and narration, not in conversation.
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No sooner had the bell rung than the students rushed out of the classroom.
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The instant the door was opened, the cat dashed outside.
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No sooner had he got home than he collapsed onto the sofa.
The verb before が早いか stays in the dictionary form even though the whole sentence is in the past.
Only for what already happened
The second half reports something that actually took place, so it is in the past. A plan, a request or an order can't follow が早いか.
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The instant the rain stopped, the children went out to play.
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Call me as soon as you get home.
- × 家に着くが早いか、電話してください → 家に着いたら、すぐ電話してください
が早いか, や否や and なり
means the same and is just as literary.
also means "as soon as", but both halves normally have the same subject; が早いか takes the same subject or a different one. In speech, the everyday way to say it is
.
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No sooner had the president entered the room than everyone stood up.
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The moment I opened the door, the cat ran out.
Lingaku