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Not quite, if all possible permutations are allowed. There are $8! = 40320$ permutations over 3 bits; 15 bits of key allows you to specify $2^{15} = 32768$ of them; hence any mapping of 15 bits will necessarily $40320-32768 = 7552$ of permutations unexpressable. It is doable if you don't allow every single permutation (e.g. allow only even permutations), ...