Find primitive polynomials for Galois field
pol = gfprimfd(m,
opt
,p)
Note:
This function performs computations in GF(p^{m}),
where p is prime. To work in GF(2^{m}), use
the |
If m
= 1, pol
= [1 1].
A polynomial is represented as a row containing the coefficients in order of ascending powers.
pol = gfprimfd(m,
searches
for one or more primitive polynomials for GF(opt
,p) p^m
),
where p
is a prime number and m
is
a positive integer. If m
= 1, pol
= [1
1]. If m
> 1,
the output pol
depends on the argument opt
as
shown in the table below. Each polynomial is represented in pol
as
a row containing the coefficients in order of ascending powers.
opt | Significance of pol | Format of pol |
---|---|---|
'min' | One primitive polynomial
for GF(p^m ) having the smallest possible number
of nonzero terms | The row vector representing the polynomial |
'max' | One primitive polynomial
for GF(p^m ) having the greatest possible number
of nonzero terms | The row vector representing the polynomial |
'all' | All primitive polynomials
for GF(p^m ) | A matrix, each row of which represents one such polynomial |
A positive integer | All primitive polynomials
for GF(p^m ) that have opt nonzero
terms | A matrix, each row of which represents one such polynomial |
The code below seeks primitive polynomials for GF(81) having
various other properties. Notice that fourterms
is
empty because no primitive polynomial for GF(81) has exactly four
nonzero terms. Also notice that fewterms
represents
a single polynomial having three terms, while threeterms
represents all of
the three-term primitive polynomials for GF(81).
p = 3; m = 4; % Work in GF(81). fewterms = gfprimfd(m,'min',p) threeterms = gfprimfd(m,3,p) fourterms = gfprimfd(m,4,p)
The output is below.
fewterms = 2 1 0 0 1 threeterms = 2 1 0 0 1 2 2 0 0 1 2 0 0 1 1 2 0 0 2 1 No primitive polynomial satisfies the given constraints. fourterms = []