Difference between revisions of "ApCoCoA:CharP.GBasisF2048"

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This command computes a Groebner basis in the field <formula> \mathbb{F}_{2048} = (/mathbb{Z}_{\setminus(2)} [x])_{\setminus(x^11 +x^3 + x^5 +x + 1)}</formula>. It uses the ApCoCoA Server and the ApCoCoALib's class [[ApCoCoALib:RingF2048|RingF2048]].  
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This command computes a Groebner basis in the field <formula> \mathbb{F}_{2048} = (\mathbb{Z}_{\setminus(2)} [x])_{\setminus(x^11 +x^3 + x^5 +x + 1)}</formula>. It uses the ApCoCoA Server and the ApCoCoALib's class [[ApCoCoALib:RingF2048|RingF2048]].  
  
 
The command's input is a an Ideal in a Ring over Z, where the elements 0,..., 2047 represent the field's elements. Details on this representation can be found [[ApCoCoA:Representation_of_finite_fields|here]]. For short, the binary representation of the number represents the coefficient vector if the polynomial in the field, e.g.  
 
The command's input is a an Ideal in a Ring over Z, where the elements 0,..., 2047 represent the field's elements. Details on this representation can be found [[ApCoCoA:Representation_of_finite_fields|here]]. For short, the binary representation of the number represents the coefficient vector if the polynomial in the field, e.g.  

Revision as of 22:03, 30 March 2008

Char2.GBasisF2048

computing a gbasis of a given ideal in <formula>\mathbb{F}_{2048}</formula>

Syntax

$char2.GBasisF2048(Ideal):List

Description

This command computes a Groebner basis in the field <formula> \mathbb{F}_{2048} = (\mathbb{Z}_{\setminus(2)} [x])_{\setminus(x^11 +x^3 + x^5 +x + 1)}</formula>. It uses the ApCoCoA Server and the ApCoCoALib's class RingF2048.

The command's input is a an Ideal in a Ring over Z, where the elements 0,..., 2047 represent the field's elements. Details on this representation can be found here. For short, the binary representation of the number represents the coefficient vector if the polynomial in the field, e.g.

<formula> 11 = 8 + 2 + 1 = 2^3 + 2^1 + 2^0</formula>

So the number <formula>11</formula> corresponds to the polynomial <formula>x^3 + x + 1</formula>.


See also

ApCoCoA:GBasis

ApCoCoA:char2.GBasisF2

ApCoCoA:char2.GBasisF4

ApCoCoA:char2.GBasisF8

ApCoCoA:char2.GBasisF16

ApCoCoA:char2.GBasisF32

ApCoCoA:char2.GBasisF64

ApCoCoA:char2.GBasisF128

ApCoCoA:char2.GBasisF256

ApCoCoA:char2.GBasisF512

ApCoCoA:char2.GBasisF1024

ApCoCoA:char2.GBasisF4096

ApCoCoA:char2.GBasisModSquares