Compacting a substitution cipher key

Question

In a substitution cipher, I recieve 3 bits and than I return another 3 bits. The key of this cipher could be a matrix represented as:

(0) 0 0 0 => 0 1 1

(1) 0 0 1 => 1 0 1

(2) 0 1 0 => 1 1 1

(3) 0 1 1 => 0 1 0

(4) 1 0 0 => 0 0 1

(5) 1 0 1 => 1 0 0

(6) 1 1 0 => 0 0 0

(7) 1 1 1 => 1 1 0

So, when we receive 000, the cipher will return 011.

When we receive 101, the cipher will return 100.

The key itself can be represented as a stream of 24 bits like this:

0 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 0

Is there a way to reduce the key to 15 bits?

I other words, can that matrix representation of 24 bits, could be represented with 15 bits?

Answer 1

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), or if you allow 16 bits of keying data.