What is a charecteristic polynomial of Gold code? I'm trying to figure out, how Gold code is generated, and what it actually its characteristic polynomial. I was watching a video and it was mentioned that the superscripts in the polynomial do not denote exponents, but just bit positions. But I'm curious, what is the reason that they are called polynomials, because it is unlikely that it this notation is chosen for its convenience.
My guess is that these are polynomials over some finite field, but I failed to find further information, as searching for Gold codes I encountered only articles on how to calculate them, instead of fundamentals.
 A: OK, firstly let's define the characteristic polynomail of an LFSR sequence, i.e., a linear recurring sequence. Given a linear recurrence of order $d,$
$$
(1)\qquad s_t = a_1 s_{t-1} + \cdots + a_d s_{n-1} \pmod 2,
$$
over $GF(2)$ its characteristic polynomial is defined as
$$
a(x)=1+a_1 x + \cdots + a_d x^d \in GF(2)[x].
$$
If this polynomial is irreducible over $GF(2)$ and has a root of maximal (multiplicative) order $2^d-1,$ in $GF(2^d)$ it is called a primitive polynomial. A primitive polynomial generates a maximal length sequence of period $2^d-1$ as long as $(s_0,\ldots,s_{d-1})$ is initialized to some nonzero binary vector and (1) is applied to generate the rest of the sequence.
Let $\alpha$ be a root of such a primitive polynomial $a(x)$ generating $(a_t).$
Let $u$ be a small positive integer such that $(u,2^d-1)=1.$ Then the polynomial $a'(x)$ having $\alpha^u$ as a root is also primitive. Let $a'(t)$ be the corrsponding sequence at some phase.
The set of Gold sequences are then
$$
\left\{ a(t) \right\} \cup \{ a'(t) \} \cup \{  a(t+k)+a'(t): 0\leq k\leq 2^d-1 \}.
$$
Edit: As pointed out in the comments, the goal is  to obtain the esentially minimum possible off peak correlation magnitude bound of $2^{(d+1)/2}+1$. One way to do this is to let $u=2^s+1,$ where $(s,d)=1.$
