From: Daniel Reischert
Newsgroups: sci.math.symbolic
Subject: Re: Large irreducible polynomials
Date: 22 Mar 1996 16:42:49 GMT
In [1] you can find the construction of
irreducible polynomials of degree m*t out of
irreducible polynomials of degree m, if t is chosen appropriately.
Taking
f1(x) = x^4 + x + 1 and
f2(x) = x^4 + x^3 + 1, which are irreducible over GF(2),
you can construct (choosing t= 3^3 * 5^4 = 16.875)
g1(x) = x^67.500 + x^16.875 + 1 and
g2(x) = x^67.500 + x^50.625 + 1,
which are also irreducible, by the theorem in [1].
These polynomials seem to be quite sparse ...
but I think, you can avoid this by switching to
g1(x+1) and g2(x+1). Right?
Another "type" of constrution (it might be the
same, I didn't check this) can be found in [2].
The polynomials constructed in this paper are
of degree 2^n, but only one polynomial per
degree is gained. Maybe there are some variations
of this procedure?
Daniel Reischert, Bonn, Germany, daniel@cs.bonn.edu
[1] R. Lidl, H. Niederreiter, "Introduction to Finite Fields and
Their Applications".
Encyklopedia of Mathematic and its Applications,
Section Algebra, Vol.20, Addison-Wesley,
Reading, Massachusetts (1983), pp. 88--89
[2] D. G. Cantor, "On Arithmetical Algorithms over Finite Fields",
Journal of Combinatorial Theory A 50 (1989), 285--300.