Let's suppose that I send the following sequence of bits:
$x = [1, 0]$
I use a 7 length Barker code ($b = [−1,−1,−1,1,1,−1,1]$) in a BPSK modulation scheme, resulting in the following sequence:
$x_B =\\ [−1,−1,−1,\:\:1,\:\:1,−1,\:\:1\\ \:\:\:1,\:\:\:1,\:\:1,-1,-1,\:\:\:1,-1]$
We send it, through a AWGN channel, and after decoding there are some errors in the sequence: $y_B =\\ [\:\:1,−1,\:\:1,\:\:1,\:\:1,−1,\:\:1\\ \:\:\:1,-1,\:\:1,-1,-1,\:\:\:1,-1]$
How would I decode this sequence?
Under the white Gaussian noise assumption, the optimal decoder would be to minimize each window of $7$ bits, that is take you first 7 symbols $$s_1 = [\:\:1,−1,\:\:1,\:\:1,\:\:1,−1,\:\:1]$$ and compute two metrics $$c_0 = \Vert s_1 - b_0 \Vert_2^2 = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 = 20$$ $$c_1 = \Vert s_1 - b_1 \Vert_2^2 = 2^2 + 2^2 = 4$$ where $b_1 = [−1,−1,−1,1,1,−1,1]$ is the Barker code corresponding to $1$ and $b_0 = [1,1,1,-1,-1,1,-1]$ is the Barker code corresponding to $0$. So if $c_0 < c_1$ then the first received bits is $0$, else $1$.
Next, grab the other $7$ symbols $$s_2 = [1,-1,1,-1,-1,1,-1]$$ and do the same thing.
PS: You could have also computed a Hamming distance as the criterion chooses the minimum number of errors.
