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Cell[CellGroupData[{
Cell["Hadamard Search Package - Readme", "Title"],
Cell["\<\
Victor Alvarez Solano, Jose Andres Armario Sampalo, Maria Dolores \
Frau Garcia and Pedro Real Jurado
Departamento de Matematica Aplicada I, University of Seville (Spain).
20 August 2006\
\>", "Subtitle"],
Cell["valvarez@us.es", "Subsubtitle"],
Cell[TextData[{
"Determining Hadamard matrices at a desired dimension is a very difficult \
task. Furthermore, the Hadamard conjecture about the existence of Hadamard \
matrices at every dimension multiple of 4 is still open. A new insight on the \
subject was provided by the tandem De Launey-Horadam in the early 90s, in \
terms of ",
StyleBox["cocyclic",
FontSlant->"Italic"],
" Hadamard matrices. The term cocyclic refers to cocycles coming from the \
second group of cohomology. The main difficulty in this approximation is how \
to explicitly construct a basis for 2-cocycles. This question is \
significantly skipped with the use of the so-called ",
StyleBox["homologial reduction method",
FontSlant->"Italic"],
", provided a ",
StyleBox["homological model",
FontSlant->"Italic"],
" for the given group is known. This package provides a means of explicitly \
constructing a basis for 2-cocycles on G from a homological model hG for G. \
Depending on the choice of the user, an exhaustive (which is only recommended \
for low dimensions) or heuristic search for cocyclic Hadamard matrices over G \
is then developed."
}], "Text"],
Cell[CellGroupData[{
Cell["Loading the package HadamardSearch", "Subsection"],
Cell[TextData[{
"The package `HadamardSearch` may be placed in folder: ",
Cell[BoxData[
\("/Applications/Mathematica 5.2.app/AddOns/Applications"\)],
"Output"],
"/.\nI use the Get request and select initialization as a cell property \
such as:"
}], "Text"],
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Cell["For questions or comments please email me at valvarez@us.es", "Text"]
}, Open ]],
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Cell["Examples to use the package and possible tests", "Section"],
Cell["\<\
All the calculations about the homological model have been provided \
by the package \"HomologyIteratedGroups\" which has been developed by the \
author as well.\
\>", "Text"],
Cell["\<\
Let consider the dihedral group D_4t = Z_2t x_chi Z_2, for the \
dihedral action chi:Z_2 x Z_2t--\[Rule]Z_2t given by chi[1,a]= 2t-a and _a_ \
otherwise. We consider the following ordering in D_4t: {0, \
0},{0,1},{1,0},{1,1},{2,0},{2,1},...,{2t-1,0},{2t-1,1}. Attending to this \
ordering, a matrix PR representing the group law in D_4t for t=4 is given by \
\
\>", "Text"],
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Cell["\<\
Assume that a basis for hG_1 is {{1,0},{0,1}}, a basis for hG_2 is \
{{2,0},{1,1},{0,2}} and a basis for hG_3 is {{3,0},{2,1},{1,2},{0,3}}. In \
these circumstances, a matrix representing F1 is \
\>", "Text"],
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Cell["\<\
Now we may proceed to develope an exahustive search (this is only \
feasible for groups of low order, up to order 28).\
\>", "Text"],
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*)
(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)