The North Africans were eliminated on away goals following 2-1 extra-time defeat on Tuesday in Blida, with Cameroon attacker Karl Toko Ekambi netted the game-winning goal in the 124th minutes.
In a statement, the federation claimed it had failed an appeal with FIFA, allerging "scandalous arbitration"that _distorted the results".
When the score was 1-0 to Cameron in the second half Algeria attacker Islam slimani had a goal ruled out for offside, and slim and then had another effort thrown out in the first part of extra time.
The desert foxes did equalised on the night and went ahead to 2-1 on aggregate thanks to Ahmed Touba's 118th minute header, but Toko Ekambi Scored deep in added time to draw the Thai at 2-2 and send the central Africans to Qatar on away goals.
"The FAF is determined to use all legally permitted means to have its rights restored and to replay the match and the conditions guaranteeing the honesty and partiality of the referee,"a statement on the FAF website said.
"The FAF also request the opening of an investigation by FIFA bodies to shed light on the arbitration of the Algeria-Cameroon match."
The draw for the group stage of the world cup is scheduled to take place on Friday.
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