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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.




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