I find confusing some examples I have seen. Maybe you can help me to determine what is going on with them.
A Generalized Feedback Shift Register (GFSR) sequence defines a sequence $\{W_{i}\}$ satisfying the equation
$$W_{k+p}=c_{0}W_{k}\bigoplus c_{1}W_{k+1}\bigoplus...\bigoplus c_{p-1}W_{k+p-1} \qquad \qquad (1)$$
where $\bigoplus$ is the binary exclusive-or operation.
If the polynomial $f(x)=c_{0}+c_{1}x+c_{2}x^{2}+...+c_{p-1}x^{p-1}+x^{p}$ is a primitive polynomial over $GF(2)$, then the sequence $\{W_{i}\}$ will have maximal sequence $2^{p}-1$.
Example 1: Let's consider the trinomial $1+x+x^{4}$ and a bit sequence $1, 0, 1, 0$. For the polynomial we have $c_{0}=1$, $c_{1}=1$, $c_{2}=0$, $c_{3}=0$ and $p=4$. Therefore, the equation $(1)$ should be $W_{k+4}=W_{k}\bigoplus W_{k+1}$. According to this, we calculate the values for $W_{5}, W_{6},...$ etc (since we already know that $W_{1}=1, W_{2}=0, W_{3}=1, W_{4}=0$).
This procedure generates the following sequence
$$1,0,1,0,1,1,1,1,0,0,0,1,0,0,1$$
Then the example takes 4 bit chunks (changing to decimal representation):
$1010=10, 1111=15, 0001=1, 0011=3, 1010+1111=0101=5, 0001+0011=0010=2$ and so on. So a '4-wise decimation' using the recurrence yields the numbers
$$10, 15, 1, 3, 5, 2, ...$$
Is this a standard way to generate a bigger sequence?
Example 2:
By the using the bit stream from the trinomial $1+x+x^{4}$ and the starting sequence $1,0,1,0$, and... forming 4-bit words by putting the bits into a fixed binary position with a delay of 3 between binary positions, we have $$1010=10, 1110=14, 0011=3, 0101=5, 1111=15, 0001 = 1, 0010=2, 0111=7,...$$
Well, both examples are dealing with exactly the same problem. However, they lead to different sequences. I don't even know how the second example generates its sequence (it looks like it is taking the first bit sequence $1, 0, 1, 0$ and applying the binary exclusive-or operation for the first two terms $1 \bigoplus 0 = 1$ which is the first term of the following bit sequence, then take the second and third terms $0 \bigoplus 1=1$ which is the second term of the new sequence and so on). However, I don't know how it gets the last term. Such a pattern works for all the sequences of the Example 2, which makes me thing that I'm not seeing the full picture.
Ideas?