The country's disaster management agency says hundreds of buildings were damaged by Saturday's tsunami.
It says the possible cause of the tsunami was undersea landslides after the Krakatoa volcano erupted.
The strait, between the islands of Java and Sumatra, connects the Java Sea to the Indian Ocean.
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The disaster management agency warned that the death toll after the tsunami, that struck at about 21:30 local time (14:30GMT) on Saturday, is likely to rise further.
Deaths have been reported in the Pandeglang, South Lampung and Serang regions.
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Among the areas hit was the popular Tanjung Lesung beach resort in west Java. There was no warning of the advancing wave.
Footage shared on social media showed a large wave crashing into a venue in the resort, where a popular Indonesian rock band, Seventeen, was performing. Members of the band are seen being swept away as the wave destroys the stage.
In an Instagram video, singer Riefian Fajarsyah, said the band's bassist and road manager had died, and that three other band members and his own wife were missing.
The country's Red Cross said it was on the scene and searching for victims in the rubble of one collapsed building.
'There were two waves'
Oystein Lund Andersen, Norwegian volcano photographer, Anyer Beach in West Java
I was on the beach. I was alone, my family were sleeping in a room.
I was trying to photograph the erupting Krakatoa volcano.
Earlier in the evening, there was quite heavy eruption activity. But just prior to the waves hitting the beach, there was no activity at all. It was just dark out there.
And suddenly I saw this wave coming, and I had to run.
There were two waves. The first wave wasn't that strong - I could run from it.
I ran straight to the hotel, where my wife and my son were sleeping.
And I woke them up... and I heard a bigger wave coming. I looked out of the window when the second wave hit. It was much bigger.
The wave passed the hotel. Cars were pushed off the road.
We and other people at the hotel went straight to the forest (on higher ground) next to the hotel. And we're still up on the hill now.
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