Quelle Util.xml Sprache: XML

<Chapter Label="ch:Utils">
<Heading>Utilities from the <Package>StandardFF</Package> package</Heading>

<Section Label="sec:SimpleRand">
<Heading>A simple bijection on a range</Heading>

<#Include Label="StandardAffineShift">

</Section>

<Section Label="sec:SEBasis">
<Heading>Finding linear combinations</Heading>

<#Include Label="FindLinearCombination">

</Section>

<Section Label="sec:IsIrred">
<Heading>Irreducibility over finite fields</Heading>

<#Include Label="IsIrreducibleCoeffList">

</Section>

<Section Label="sec:Conway">
<Heading>Connection to Conway polynomials</Heading>

<#Include Label="FindConjugateZeroes">
<#Include Label="ZeroesConway">
<#Include Label="SteinitzPairConwayGenerator">

</Section>

<Section Label="sec:DLog">
<Heading>Discrete logarithms</Heading>

<#Include Label="DLog">

</Section>

<Section Label="sec:BM">
<Heading>Minimal polynomials of sequences</Heading>

<#Include Label="InvModCoeffs">

<#Include Label="BerlekampMassey">

</Section>

<Section Label="sec:BrauerChar">
<Heading>Brauer characters with respect to different lifts</Heading>

Let  <M>G</M> be  a  finite group,  <M>g  \in G</M>,  and  <M>\rho: G  \to
GL(d,p^n)</M>be a representation over a finite field. The Brauer character
value  <M>\chi(g)</M>  of  <M>\rho</M>  at  <M>g</M>  is  defined  as  the
sum  of the  eigenvalues of  <M>\rho(g)</M>  in the  algebraic closure  of
<M>\mathbb{F}_p</M> lifted to complex roots of unity.
<P/>
The lift  used by <Ref  BookName="Reference" Oper="BrauerCharacterValue"/>
and in the computation of  many Brauer character tables (available through
the <Package>CTblLib</Package>  package) is defined by  Conway polynomials
(see <Ref BookName="Reference" Func="ConwayPolynomial"/>): They define the
primitive root  <C>Z(q)</C> in <C>GF(q)</C>  which is mapped  to <M>\exp(2
\pi i / (q-1))</M> (that is <C>E(q-1)</C> in &GAP;).
<P/>
Another     lift      is     defined     by     the      function     <Ref
Func="StandardCyclicGenerator"/>   provided   by   this   package.   Here,
<C>StandardCyclicGenerator(F, m)</C> is mapped to <M>\exp(2 \pi i / m)</M>
(that is <C>E(m)</C> in &GAP;).
<P/>
The following function translates between these two lifts.

<#Include Label="StandardValuesBrauerCharacter">

The inverse of a lift is used to reduce character values in characteristic
<M>0</M>  modulo  a   prime  <M>p</M>.  Choosing  a   lift  is  equivalent
to  choosing  a  <M>p</M>-modular  system. &GAP;  has  the  function  <Ref
BookName="Reference" Attr="FrobeniusCharacterValue"/>  which computes this
reduction with respect to the lift defined by Conway polynomials.
<P/>
Here is the corresponding function with respect to the lift constructed in
this package.

<#Include Label="FrobeniusCharacterValues">

</Section>

<Section Label="sec:FactorData">
<Heading>Known factorizations of multiplicative group orders</Heading>

<#Include Label="CANFACT">

</Section>

<Section Label="sec:Tests">
<Heading>Some loops for  <Package>StandardFF</Package></Heading>

<#Include Label="TestLoops">

</Section>

<Section Label="sec:nondoc">
<Heading>Undocumented features</Heading>
We mention some features of this package which may be temporary, vanish or
changed.
<P/>
A   directory  <F>ntl</F>   contains  some   simple  standalone   programs
which  use  the  library  NTL  <Cite  Key="NTL"/>.  There  is  a  function
<C>StandardIrreducibleCoeffListNTL(K, d, a)</C> which  can be used instead
of  <C>StandardIrreducibleCoeffListNTL(K, d,  a)</C>  when  <C>K</C> is  a
prime field.  This gives a  good speedup for  not too small  <C>d</C>, say
<C>d</C> <M>>500</M>.
</Section>

</Chapter>

quality98%

¤ Dauer der Verarbeitung: 0.1 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.