I found two definitions Reed-Muller codes being used in literature. More specifically for any $n \in \mathbb{Z}^+$ and $1 \leq d \leq n$ we define the set $RM(d,n)$ in two possible ways,
1. $RM(d,n) = \{ f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2 \vert $f$ \text{ is a polynomial of degree at most } $d$\} $
2. $RM(d,n) = \{ f: \mathbb{F}_2^n \rightarrow \mathbb{F}_2 \text{ s.t} f(x) = (-1)^{P(x)} \vert $P$ \text{ is a polynomial of degree at most } $d$ \text{and with integer coefficients of magnitude at most }d \}$
- Are these the same?
- In the first definition how is "degree" defined? Is $f$ assumed to be a sum over monomials where each literal occurs with power $1$ and then the final answer of the polynomial is evaluated $mod$ $2$ or is $f$ to be thought of as a generic degree $d$ polynomial such that when its input is restricted to $\mathbb{F}_2^n$ its output gets restricted to $\mathbb{F}_2$?