orthogonal binary sequences

Question

How to show that two binary sequences are orthogonal?
For an example verify whether [0110001] and [0011101] are orthogonal

Answer 1

Depending on your specific inner product, I will assume that the inner product is given by $\langle u,v\rangle = \sum(u)_i(v)_i$ and that we are working in $\mathbb{F}_2^7$ as our vector space (i.e. the space of binary sequences of length seven).

So, the inner product between the two is given by multiplying each position together and then adding:

$\begin{array}{llllllll}&\color{red}{0}&\color{brown}{1}&\color{orange}{1}&\color{green}{0}&\color{blue}{0}&\color{purple}{0}&1\\ \times & \color{red}{0}&\color{brown}{0}&\color{orange}{1}&\color{green}{1}&\color{blue}{1}&\color{purple}{0}&1\\ \hline &\color{red}{0}+&\color{brown}{0}+&\color{orange}{1}+&\color{green}{0}+&\color{blue}{0}+&\color{purple}{0}+&1 \end{array}$

In the context of $\mathbb{F}_2$, you have $1+1=0$.

Since $\langle u,v\rangle = 0$, these two sequences are indeed orthogonal.