I need some clarification about what the following thing is called (it's commonly used to describe linear-feedback shift registers), and what a good computer program identifier would be that describes it.
The set $p(x)$ of polynomials in $x$ with coefficients in GF(2), modulo a characteristic polynomial $p_c(x)$ with coefficients in GF(2).
I'm not a mathematician, so I always get this wrong. Is it a ring of polynomials over the field GF(2)? How do you work the characteristic polynomial into the terminology? ($p_c(x)$ may or may not be irreducible; I work with both cases.)
And what would be a good computer program identifier to represent the set of polynomials for a specific characteristic polynomial $p_c(x)$? GF2PolynomialRing? GF2PolynomialModRing?
If the polynomial is irreducible of degree $n$, it's what mathematicians would call the finite field of order $2^n$ and what people in other areas might call a Galois field $\text{GF}(2^n)$.
GF(2, n)looks good to me. It is a cool and surprising fact that this finite field does not depend on the choice of polynomial, as long as it is irreducible of degree $n$, which is why it's not mentioned in the notation.(For the purpose of describing LFSRs you need a stronger condition than irreducibility.)