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#Japt, 29 11 bytes

Uo ®¤¬r@X^Y 

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Outputs directly as an array, which saves about 4 bytes.

###Ungolfed and explanation

Uo ® ¤ ¬ r@ X^Y Uo mZ{Zs2 q rXY{X^Y}} // Implicit: U = input number Uo // Create an array of integers in the range `[0, U)`. mZ{Zs2 // Map each item Z in this range to Z.toString(2), q rXY{ // split into chars, and reduced by X^Y}} // XORing. // This returns (number of 1s in the binary string) % 2. // Implicit: output last expression 

Edit: You can now use the following 8-byte code (not valid; feature published after this challenge):

Uo ®¤¬r^