What is the name of a matrix displaying all possible combinations of a set (without repetition)? Suppose you have 4 letters: A, B, C, D.
Suppose you want each possible combinations of these letters without repetition: i.e.
{A}, {B}, {C}, {D}, {A,B}, {A,C}, {A,D}, {B,C}, {B,D}, {C,D}, {A,B,C}, {A,B,D}, {A,C,D}, {B,C,D}, {A,B,C,D}
Each combination can be represented by a vector, where the i-th element of the vector takes 1 if the letter is included in the combination, and 0 otherwise. For instance, combinations {A}, {B}, {C}, {D} would be represented respectively as
{1, 0, 0, 0}
{0, 1, 0, 0}
{0, 0, 1, 0}
{0, 0, 0, 1}
By iterating this process, I can create a matrix of this form.
$$
\begin{array}{rrrr}
  \hline
A & B & C & D \\ 
  \hline
1 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
1 & 1 & 0 & 0 \\ 
1 & 0 & 1 & 0 \\ 
1 & 0 & 0 & 1 \\ 
0 & 1 & 1 & 0 \\ 
0 & 1 & 0 & 1 \\ 
0 & 0 & 1 & 1 \\ 
1 & 1 & 1 & 0 \\ 
1 & 1 & 0 & 1 \\ 
1 & 0 & 1 & 1 \\ 
0 & 1 & 1 & 1 \\ 
1 & 1 & 1 & 1 \\ 
   \hline
\end{array}
$$
My question is: does this matrix have any known name? Alternatively, is there any known method to formalize how this matrix is produced, so that I can use a general formula to refer to any given row of the matrix?
Note that I am not interested in the order of the rows of the matrix, this can be also random.
 A: Another way to think about the rows of this matrix is that (after a suitable reordering)  they are the binary representations of the numbers $1, 2, \dots, 2^4 - 1$. If we included the empty set, its row (which would be all zeroes) would be the binary representation of $0$. Various ways to get the binary representations of numbers are your best way to get the $i^{\text{th}}$ row of the matrix in terms of $i$.
If you just want to generate all the rows, your best bet is a recursive strategy. To get the $2^n$ rows for the $n$-column case (including the zero row), take two copies of the $(n-1)$-column matrix. Add a $0$ in the $n^{\text{th}}$ column of the first copy, and a $1$ in the $n^{\text{th}}$ column of the second copy.
Most applications don't actually treat this table as a matrix. But you see it in error-correcting codes: if we take the transpose (so that we have $n$ rows and $2^n-1$ columns) then the result is the parity check matrix of the Hamming code, or equivalently the generator matrix of the Hadamard code. (For these applications, the order of your rows/their columns is not terribly important either, though you want to pick an order and stick to it, so you'll see various permutations.)
