"Name" for this order relation of binary sequence

Question

I am trying to define a Matrix $M_n$ of dimension $2^n \times n$ where each row corresponds to a element of the set $\{0,1\}^n$. To avoid ambiguity, I using an order relation over the set $\{0,1\}^n$ to sort the rows of $M_n$.

Currently, I am using the following order, illustrate here by $M_3$:

$$ M_3 = \begin{bmatrix} 0&0&0\\ 1&0&0\\ 0&1&0\\ 0&0&1\\ 1&1&0\\ 1&0&1\\ 0&1&1\\ 1&1&1\\ \end{bmatrix}$$

I don't know how to refer to this order and I don't know how to explain the pattern used above in a simple way for a general case.

I know that I could sort the rows by the lexicographic order or by transforming the binary into an integer, but, for me, the order above it is a more natural way to order binaries sequences with a fixed length.

Is there a "special" name for this order relation presented in the example? Or is there a better way to "sort" the elements of the set $\{0,1\}^n$?

Alternatively, can someone help me to define the presented order in a clear and precise way?

EDIT:

better way in the sense of a) been easier to explain/define or b) that I can present through a citation.

Answer 1

I don't know whether this order actually has a name, so I'll aim for being descriptive. Something like this would probably be fine:

Order the rows by the number of 1's. Then order each subset in reverse lexicographic order / in descending order

As for whether there is a "better" way, that depends entirely on what you need it for. If it's just for you to make sure you remember them all as you write the list, the one you feel the most comfortable with is best.