Re: Irreducible Polynomial over GF(2)
- To: mathgroup at smc.vnet.net
- Subject: [mg38734] Re: [mg38716] Irreducible Polynomial over GF(2)
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Tue, 7 Jan 2003 07:26:57 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Is this what you mean: <<Algebra`FiniteFields` FieldIrreducible[GF[2^50],x] 1 + x^46 + x^47 + x^48 + x^50 ? With best regards Andrzej Kozlowski Yokohama, Japan http://www.mimuw.edu.pl/~akoz/ http://platon.c.u-tokyo.ac.jp/andrzej/ On Tuesday, January 7, 2003, at 12:27 AM, Flip wrote: > Hello Andrzej, > > I am assuming that the form is not known, only that we want a 50th > degree irreducible. > > The solution I posted, I saw in a NG and the person was able to > determine the form and then generate the solutions you give below. > > Is there anyway to have Mathematica determine the form and generate the > solutions? > > Thank you, Flip > > --- Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: >> I am not sure xactly what you are asking for. However, it is easy to >> replicate with Mathematica your computation of the number of >> irreducible polynomials of the form: >> x^50 +x^m + x^n + x^k+ 1 >> >> In[1]:= >> testQ[{m_, n_, k_}] := >> Head[Factor[x^50 + x^m + x^n + x^k + 1, Modulus -> 2]] === >> Plus >> >> In[2]:= >> t = Flatten[Table[{m, n, k}, {m, 49}, {n, m - 1}, >> {k, n - 1}], 2]; >> >> In[3]:= >> Timing[ans = Select[t, testQ]; ] >> >> Out[3]= >> {75.39*Second, Null} >> >> In[4]:= >> Length[ans] >> >> Out[4]= >> 1421 >> >> >> Andrzej Kozlowski >> Yokohama, Japan >> http://www.mimuw.edu.pl/~akoz/ >> http://platon.c.u-tokyo.ac.jp/andrzej/ >> >> >> >> On Monday, January 6, 2003, at 05:44 PM, flip wrote: >> >>> 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. >>> >>> >>> >>> >>> >> Andrzej Kozlowski >> Yokohama, Japan >> http://www.mimuw.edu.pl/~akoz/ >> http://platon.c.u-tokyo.ac.jp/andrzej/ > > _____________________________________________________________ > Visit our web directory and reference library at http://safebunch.com. > Get FREE email for life at ---> http://safebunch.com > > _____________________________________________________________ > Select your own custom email address for FREE! Get you at yourchoice.com > w/No Ads, 6MB, POP & more! > http://www.everyone.net/selectmail?campaign=tag > >