Quantum circuit that puts qubits in "equal" superposition

I would like to build a circuit that takes multiple qubit system states and puts them in a superposition with equal amplitudes.

For example:
Let’s consider a 4 qubit system. I would like the circuit to start with an input of $left| 0000 right>$ and end with the following superposition: $$frac{1}{sqrt{4}}left(left| 1000 right> + left| 1010 right> + left| 1001 right> + left| 1100 right>right).$$

I would like for the choice of states and number of states to be arbitrary. I.e. that I can easily perform the same circuit to generate

$$frac{1}{sqrt{3}}left(left| 1000 right> + left| 1011 right> + left| 1101 right>right),$$

or

$$frac{1}{sqrt{5}}left(left| 1000 right> + left| 1010 right> + left| 1001 right> + left| 1100 right> + left| 1111 right>right).$$

Can anyone help me with this? Any help is appreciated!

EDIT: I have tried creating a matrix representation of such an operator, and I have found one that does what I need it to do, it maps the $left|00right>$ to an arbitrary equally-distributed superposition of the basis vectors of my choosing. Here is the matrix I found that acts on 2 qubits to create such a state:

$$
U = begin{pmatrix}
1/sqrt{3} & 0 & 0 & 0
1/sqrt{3} & 0 & 0 & 0
1/sqrt{3} & 0 & 0 & 0
0 & 0 & 0 & 0
end{pmatrix}.
$$

This gives:

$$
Uleft|00right>= begin{pmatrix}
1/sqrt{3} & 0 & 0 & 0
1/sqrt{3} & 0 & 0 & 0
1/sqrt{3} & 0 & 0 & 0
0 & 0 & 0 & 0
end{pmatrix}
begin{pmatrix}
1
0
0
0
end{pmatrix} =
frac{1}{sqrt{3}} begin{pmatrix}
1
1
1
0
end{pmatrix} = frac{1}{sqrt{3}}left( left|00right> + left|01right> + left|10right> right).
$$

The idea is that the first column of the matrix would consist of $N$ non-zero elements, each element contrbuting one basis state to the final superposition, and each element being equal to $frac{1}{sqrt{N}}$.

The problem with this matrix, however, is that it is not unitary. Would it be possible to modify this matrix in such a wat that it still has the same result on the $|00>$ state vector, but that it is unitary? This would solve my problem completely.