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.