This is kind of out of the blue. But a slide I'm reading describes some requirement on the construction of a parity matrix that I just don't understand - and was hoping maybe it is abstract enough for someone here to get:
"The equation $xH^{T}=0$ can be considered as a condition that a number of columns in H add up to 0. If all columns in H are different and unequal to the all-zero column, then the weight of x in order to satisfy this equation is either equal to zero or at least equal to three"
The context is parity-check matrices, and the term weight refers to the number of places in which a vector differs from the zero vector if that helps.
The matrix $H$ has dimensions $(2^{m} - 1) \times (m \geq 2)$ if that helps. All columns are also nonzero. But I'm not sure why that is either.
My confusion is the "at least equal to three" part. I can see how that if $xH^{T} = 0$ is a condition, than $x=0$ vectors solves this.
