PRNG that uses LFSR over the field GF(q): database of irreducible polynomials, period of random sequence

The task is to make a program that simulates pseudo-random sequence generator that uses LFSR, that can be described by recurrence over the field GF(q)

Y(n+1)=F * Y(n),

where F is square characteristic matrix of size k of LFSR; Y(n) is the vector of n-th state of LFSR. (That is, as I see, in case of LFSR, is the same as linear reccurence y_n+k = a_k-1*y_n+k-1+a_k-2*y_n+k-2+...+a₀*y₀, where Y(n)=(y_n, y_n+1, ... y_n+k-1)).

By the task, he program should contains the database of irreducible polynomials, should allow to calculate sequences of LFSR states and map them to integer and real numbers, to define the period of the matrix F and the initial state Y(0).

1. What do we need the database for? As I guess, the database should contain exactly one irreducible polynomial f(x) of degree n over the field F_p (we can take any of such polynomials) for each q=pⁿ, where q is number of elements of field GF(q). When we generate sequence with LFSR over the field GF(q) for certain q, first we take the appropriate polynomial from database and use it to construct the addition and multiplication tables for GF(q).

2. As I see, we can find the period of sequence for some special cases of recurrence is ("Introduction to Finite Fields and Their Applications", Rudolf Lidl, Harald Niederreiter), but not for any given initial vector and recurrence. So how we can calculate it?