Let $C$ be the ASCII code. So, $$C:=\{ (x_1,...,x_8) \in \Bbb{Z}_2^8: x_1+...+x_8=0 \} \subseteq \Bbb{Z}_2 ^8 $$ and it's not difficult to show that it's a linear code.
I found on a book (without any further explanation) that this linear code has dimension $7$. So it is a $[8,7]$-code.
But how do we conclude that the dimension is $7$? Can we find a basis for this subspace?
Thank you.
Note that the basis $$(1,1,0,0,0,0,0,0)\\(0,1,1,0,0,0,0,0)\\(0,0,1,1,0,0,0,0)\\(0,0,0,1,1,0,0,0)\\(0,0,0,0,1,1,0,0)\\(0,0,0,0,0,1,1,0)\\(0,0,0,0,0,0,1,1)\\$$is sufficient for describing $C$ and they are linearly independent (are calculations are mod 2).