Irreducible Polynomial over GF(2)
- To: mathgroup at smc.vnet.net
- Subject: [mg38716] Irreducible Polynomial over GF(2)
- From: "flip" <flip_alpha at safebunch.com>
- Date: Mon, 6 Jan 2003 03:44:30 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Hi All, I saw this posted in another NG and was wondering if this can be done in Mathematica? > > > I need an irreducible monic polynomial over Z_2 with degree 50 which > > > should also contain only one another power of variable x. So I'm > > > interested to know, for example, if x^50 + x^3 + 1 or x^50 + x + 1 or > > > something like them is irreducible in that field. > > > > There is no such polynomial. > > See EIS A073571 here: http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?An um=A073571 > Very interesting link. > Well...I saw that I can't use trinomials so I have to change my target: > I need a polynomial which should contain as few elements as possible. > For example, I truly hope that some expression such as: x^50 + x^m + x^n > + 1 would be irreducible over GF(2) for some m and n. Someone posted this reply ... You are unlikely to find a list anywhere of the 22517997465744 irreducible polynomials of degree 50 over GF(2). :-) No polynomial of the form x^50 + x^m + x^n + 1 is irreducible over GF(2) since 1 is a root. *** Can Mathematica do this? *** However, using another system I have found many of the form x^50 + x^m + x^n + x^k+ 1 which are irreducible. In fact there are 1421 of them and 826 of these are primitive. Here are the first 10 primitive polynomials of this type. (I print only m,n,k): 1, 2, 16 1, 2, 42 1, 4, 29 1, 4, 36 1, 5, 30 1, 5, 37 1, 6, 47 1, 7, 20 1, 8, 49 1, 10, 47 Thank you, Flip Remove "_alpha" to email me.