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(*CacheID: 232*)
(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[ 471783, 10353]*)
(*NotebookOutlinePosition[ 472572, 10381]*)
(* CellTagsIndexPosition[ 472528, 10377]*)
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Notebook[{
Cell[CellGroupData[{
Cell["Hadamard Search Package", "Title",
TextAlignment->Center],
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)
19 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["Reference", "Section"],
Cell[CellGroupData[{
Cell["Title", "Subsubsection"],
Cell[TextData[StyleBox["Hadamard Search",
FontSlant->"Italic"]], "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Author", "Subsubsection"],
Cell["\<\
Victor Alvarez Solano, Jose Andres Armario Sampalo, Maria Dolores \
Frau Garcia and Pedro Real Jurado\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Summary", "Subsubsection"],
Cell["\<\
This package is designed to find out some cocyclic Hadamard \
matrices over a finite group G, provided a homological model for G is \
known.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Copyright", "Subsubsection"],
Cell["\<\
Rights reserved by Victor Alvarez Solano et al, Departamento de \
Matematica Aplicada I, University of Seville (Spain).\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Notebook Version", "Subsubsection"],
Cell["2.0", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[{
StyleBox["Mathematica",
FontSlant->"Italic"],
" Version"
}], "Subsubsection"],
Cell["4.0", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["History", "Subsubsection"],
Cell["\<\
This package is the implementation in practise of the homological \
reduction method for constructing a basis for 2-cocycles over a finite group \
for which a homological model is known. Furthermore, a routine for developing \
a heuristic search (in terms of a genetic algorithm) is also included. This \
tool is very useful when exhaustive search is not possible (usually for \
groups of order equal to or greater than 28).\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Keywords", "Subsubsection"],
Cell["\<\
Hadamard matrix, cocyclic Hadamard matrix, homological model, \
homological reduction method.\
\>", "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Limitation", "Subsubsection"],
Cell[TextData[StyleBox["This package works only on finite groups for which \
some homological models are known.",
FontSlant->"Italic"]], "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Discussion", "Subsubsection"],
Cell[TextData[StyleBox["References:\nAn algorithm for computing cocyclic \
matrices developed over some semidirect products. V. Alvarez, J.A. Armario, \
M.D. Frau and P. Real, LNCS 2227, 287--296 (2001).\nA genetic algorithm for \
cocyclic Hadamard matrices. V. Alvarez, J.A. Armario, M.D. Frau and P. Real, \
LNCS 3857, 144--153 (2006).\nCalculating cocyclic Hadamard matrices in \
Mathematica: exhaustive and heuristic searches. V. Alvarez, J.A. Armario, \
M.D. Frau and P. Real, LNCS 4151, 119--122 (2006).\nHomological reduction \
method for constructing cocyclic Hadamard matrices. V. Alvarez, J.A. Armario, \
M.D. Frau and P. Real (in preparation).",
FontSlant->"Italic"]], "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell["Requirements", "Subsubsection"],
Cell[TextData[{
StyleBox["Context`HadamardSearch`\nuses ",
FontFamily->"Courier",
FontWeight->"Bold",
FontSlant->"Italic"],
"IntegerSmithNormalForm, due to V. Alvarez et al, 2006;"
}], "Text"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["Interface", "Section",
InitializationCell->True],
Cell["\<\
This part declares the publicly visible functions, options, and \
values.\
\>", "Text",
InitializationCell->True],
Cell[CellGroupData[{
Cell["Set up the package context, including public imports", "Subsection",
InitializationCell->True],
Cell[CellGroupData[{
Cell["BeginPackage[\"`HadamardSearch`\"]", "Input",
InitializationCell->True],
Cell[BoxData[
\("Global`HadamardSearch`"\)], "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
\($ContextPath\)], "Input"],
Cell[BoxData[
\({"Global`HadamardSearch`", "System`"}\)], "Output"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
Usage messages for the exported functions and the context \
itself\
\>", "Subsection",
InitializationCell->True],
Cell[CellGroupData[{
Cell["\<\
HadamardSearch::usage=\"HadamardSearch[PR,M2,M3,F1,F2,method] \
searches for cocyclic Hadamard matrices over a finite group G of group law \
given by the square matrix PR. The notebook will assume the ordering implicit \
in PR for the elements in G. Special care must be taken so that the first \
element in G is the identity element. The search may be both exhaustive and \
heuristic, depending on whether method=1 or method !=1. The matrices M2 and \
M3 represent the differentials d2 and d3 on the homological model hG. The \
matrices F1 and F2 represent the projection maps from B_1(Z[G]) to hG_1 and \
B_2(Z[G]) to hG_2, respectively. Let denote the number of elements in G by \
_order_ (=Length[PR]). The _order_ elements in B_1(Z[G]) are ordered \
following the natural ordering induced by PR. The _order_^2 elements in \
B_2(Z[G]) are ordered as the elements of G x G, from the first row to the \
last one, from left to the right, as \
{1,1},{1,2},...,{1,_order_},{2,1},...,{2,_order_},...,{_order_,1},...{_order_,\
_order_}. Of course, the same basis for hG at each degree must be used for \
M2,M3,F1 and F2. There is an explicit example included as a comment at the \
end of the package.\"\
\>", "Input",
InitializationCell->True],
Cell[BoxData[
\("HadamardSearch[PR,M2,M3,F1,F2,method] searches for cocyclic Hadamard \
matrices over a finite group G of group law given by the square matrix PR. \
The notebook will assume the ordering implicit in PR for the elements in G. \
Special care must be taken so that the first element in G is the identity \
element. The search may be both exhaustive and heuristic, depending on \
whether method=1 or method !=1. The matrices M2 and M3 represent the \
differentials d2 and d3 on the homological model hG. The matrices F1 and F2 \
represent the projection maps from B_1(Z[G]) to hG_1 and B_2(Z[G]) to hG_2, \
respectively. Let denote the number of elements in G by _order_ \
(=Length[PR]). The _order_ elements in B_1(Z[G]) are ordered following the \
natural ordering induced by PR. The _order_^2 elements in B_2(Z[G]) are \
ordered as the elements of G x G, from the first row to the last one, from \
left to the right, as \
{1,1},{1,2},...,{1,_order_},{2,1},...,{2,_order_},...,{_order_,1},...{_order_,\
_order_}. Of course, the same basis for hG at each degree must be used for \
M2,M3,F1 and F2. There is an explicit example included as a comment at the \
end of the package."\)], "Output"]
}, Open ]]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["Implementation", "Section",
InitializationCell->True],
Cell["\<\
This part contains the actual definitions and any auxiliary \
functions that should not be visible outside.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Begin the private context (implementation part)", "Subsection",
InitializationCell->True],
Cell[CellGroupData[{
Cell["\<\
Begin[\"`Private`\"]
{$Context,$ContextPath}\
\>", "Input",
InitializationCell->True],
Cell[BoxData[
\("Global`HadamardSearch`Private`"\)], "Output"],
Cell[BoxData[
\({"Global`HadamardSearch`Private`", {"Global`HadamardSearch`",
"System`"}}\)], "Output"]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["\<\
Definition of auxiliary functions and local (static) \
variables\
\>", "Subsection",
InitializationCell->True],
Cell[BoxData[{
\( (*\[IndentingNewLine]\[IndentingNewLine]We\ use\ the\ \
IntegerSmithNormalForm\ package\ due\ to\ V . \ Alvarez\ et\ al, \
2006. \[IndentingNewLine]\[IndentingNewLine]*) \n\(M[i_, j_, t_, k_] :=
ReplacePart[IdentityMatrix[k],
t, {i, j}];\)\), "\[IndentingNewLine]",
\(\(T[i_, j_, k_] :=
Module[{m, n}, m = IdentityMatrix[k]; n = m[\([i]\)];
m[\([i]\)] = m[\([j]\)]; m[\([j]\)] = n;
m];\)\), "\[IndentingNewLine]",
\(\(Completar[C_, m_, n_] :=
Module[{k, i, j, ii}, i = m - Length[C]; k = {};
Do[k = Append[k, Table[0, {ii, 1, n}]], {j, 1, i}];
Do[k = Append[k,
Join[Table[0, {ii, 1, n - Length[C[\([1]\)]]}],
C[\([j]\)]]], {j, 1, Length[C]}];
k];\)\), "\[IndentingNewLine]",
\(\(ExtendedSmithForm[A_] :=
Module[{k, C, coli, colj, P, Q, m, n, H, i, it, control},
m = Length[A]; n = Length[A[\([1]\)]]; P = {IdentityMatrix[m]};
Q = {IdentityMatrix[n]}; C = A; i = 1;
Fin = Max[Abs[Take[C, {i, m}, {i, n}]]] >
0; \[IndentingNewLine]While[
Fin, {coli,
colj} = \(Position[
Completar[Take[Abs[C], {i, m}, {i, n}], m, n],
Min[Select[
Flatten[
Take[Abs[C], {i, m}, {i, n}]], #1 >
0 &]]]\)[\([1]\)]; P = Prepend[P, T[i, coli, m]];
Q = Append[Q, T[i, colj, n]]; it = i + 1;
C = First[P] . C . Last[Q];
control = True; \[IndentingNewLine]While[it \[LessEqual] n,
If[IntegerQ[C[\([i, it]\)]/C[\([i, i]\)]], it = it + 1,
Q = Append[Q,
M[i, it, \(-Quotient[C[\([i, it]\)], C[\([i, i]\)]]\),
n] . T[it, i, n]]; C = C . Last[Q]; control = False;
it = n + 1]\[IndentingNewLine]];
If[control,
Do[Q = Append[Q,
M[i, it, \(-C[\([i, it]\)]\)/C[\([i, i]\)], n]];
C = C . Last[Q], {it, i + 1, n}];
it = i + 1; \[IndentingNewLine]While[it \[LessEqual] m,
If[IntegerQ[C[\([it, i]\)]/C[\([i, i]\)]], it = it + 1,
P = Prepend[P,
T[it, i, m] .
M[it, i, \(-Quotient[C[\([it, i]\)],
C[\([i, i]\)]]\), m]]; C = First[P] . C;
control = False; it = m + 1]\[IndentingNewLine]];
If[control,
Do[P = Prepend[P,
M[it, i, \(-C[\([it, i]\)]\)/C[\([i, i]\)], m]];
C = First[P] . C, {it, i + 1, m}]; \[IndentingNewLine]Catch[
Do[\[IndentingNewLine]Do[
If[IntegerQ[C[\([coli, colj]\)]/C[\([i, i]\)]], Null,
P = Prepend[P, M[i, coli, 1, m]]; C = First[P] . C;
control = False; Throw[False]], {colj, i + 1,
n}], {coli, i + 1, m}]; Throw[True]];
If[\ control,
P = Prepend[P, M[i, i, Sign[C[\([i, i]\)]], m]];
C = First[P] . C; i = i + 1;
Fin = Max[Abs[Take[C, {i, m}, {i, n}]]] >
0]\[IndentingNewLine]];\[IndentingNewLine]];\
\[IndentingNewLine]]; \[IndentingNewLine]P = Fold[Dot, IdentityMatrix[m], P];
Q = Fold[Dot, IdentityMatrix[n], Q]; {C, {P,
Q}}\[IndentingNewLine]];\)\[IndentingNewLine] \
(*\[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine]*) \)}], "Input",
InitializationCell->True],
Cell[BoxData[{
\( (*\ \[IndentingNewLine]\[IndentingNewLine]Auxiliary\ functions\ for\ \
constructing\ a\ basis\ for\ representative\ 2 -
cocycles\ coming\ from\ \(\(inflation\)\(.\)\)\[IndentingNewLine]\
\[IndentingNewLine]*) \[IndentingNewLine]\(matrizdeunos[n_] :=
Table[Table[1, {j, n}], {i, n}];\)\), "\[IndentingNewLine]",
\(\(matriznegaciclica[n_] :=
Table[Join[Table[1, {j1, n + 1 - j}],
Table[\(-1\), {j2, n + 2 - j, n}]], {j,
n}];\)\), "\[IndentingNewLine]",
\(\(kronunosaizq[a_, n_] :=
Flatten[Table[
Table[Apply[Join,
Table[Table[a[\([i, k]\)], {k2, n}], {k,
Length[a[\([1]\)]]}]], {j, n}], {i, Length[a[\([1]\)]]}],
1];\)\), "\[IndentingNewLine]",
\(\(kronunosader[a_, n_] :=
Flatten[Table[
Table[Apply[Join, Table[a[\([j]\)], {k, n}]], {j,
Length[a[\([1]\)]]}], {i, n}], 1];\)\)}], "Input",
InitializationCell->True],
Cell[BoxData[
\(\(\( (*\[IndentingNewLine]Cocyclic\ Hadamard\ \(\(Test\)\(.\)\)\ \
\[IndentingNewLine]*) \)\(\[IndentingNewLine]\)\(testhadamard[
L_] := \n\t\tCatch[
Do[If[Apply[Plus, L[\([i]\)]] \[NotEqual] 0, Throw[False]], {i, 2,
Length[L]}]; Throw[True]];\)\)\)], "Input",
InitializationCell->True],
Cell[BoxData[
\(\(\( (*\ \[IndentingNewLine]Hadamard\ pointwise\ \(\(product\)\(.\)\)\ \
\[IndentingNewLine]*) \)\(\[IndentingNewLine]\)\(prodhadamard[a_, b_] :=
Table[Table[
a[\([i, j]\)]*b[\([i, j]\)], {j, Length[a[\([1]\)]]}], {i,
Length[a[\([1]\)]]}];\)\)\)], "Input",
InitializationCell->True]
}, Open ]],
Cell[CellGroupData[{
Cell["Main program", "Subsection",
InitializationCell->True],
Cell[BoxData[
\(\(\( (*\[IndentingNewLine]\[IndentingNewLine]Main\ program\
\[IndentingNewLine]\[IndentingNewLine]*) \)\(\[IndentingNewLine]\)\(\
HadamardSearch[PR_, M2_, M3_, F1_, F2_, method_] :=
Module[{}, \[IndentingNewLine] (*\ \[IndentingNewLine]\
\[IndentingNewLine]Calculating\ H1\ and\ \(\(H2\)\(.\)\)\ \[IndentingNewLine]\
\[IndentingNewLine]*) \[IndentingNewLine] (*\
b1\ and\ b2\ denote\ the\ number\ of\ generators\ in\ the\ \
homological\ model\ at\ degrees\ 1\ and\ 2, \ \(\(respectively\)\(.\)\)\ \
*) \[IndentingNewLine]fnsa1 = ExtendedSmithForm[M2];
b1 = Length[M2[\([1]\)]];
b2 = Length[M2]; \[IndentingNewLine]abelianizado =
Select[Table[
fnsa1[\([1, ji, ji]\)], {ji, b1}], #1 >
0 &]; \[IndentingNewLine]b1 = b2;
b2 = Length[M3]; \[IndentingNewLine]fnsa2 =
ExtendedSmithForm[M3]; \[IndentingNewLine]homologia =
Select[Table[
fnsa2[\([1, ji, ji]\)], {ji, b1}], #1 >
0 &]; \[IndentingNewLine] (*\[IndentingNewLine]\ \
\[IndentingNewLine]Constructing\ a\ basis\ _base2cociclos _\ for\ normalized\
\ 2 - \(\(cocycles\)\(.\)\)\ \[IndentingNewLine]\[IndentingNewLine]\
*) \[IndentingNewLine]base2cociclos = {}; \
\[IndentingNewLine]Print["\"]; \
\[IndentingNewLine] (*\ \[IndentingNewLine]\[IndentingNewLine]Constructing\ a\
\ basis\ for\ 2 - \(\(coboundaries\)\(.\)\)\ \[IndentingNewLine]\
\[IndentingNewLine]*) \[IndentingNewLine]orden =
Length[PR]; \[IndentingNewLine]cobordes =
Table[Apply[Join,
Table[Table[
Mod[KroneckerDelta[i, k] + KroneckerDelta[i, j] +
KroneckerDelta[i, PR[\([k, j]\)]], 2], {j,
orden}], {k, orden}]], {i, 2,
orden}]; \[IndentingNewLine]m = cobordes;
gen = {}; \[IndentingNewLine]i = 0; \[IndentingNewLine]While[
i < orden - 1, \[IndentingNewLine]n =
Select[Range[i + 1, orden - 1],
m[\([#1]\)] \[NotEqual] Table[0, {k, orden^2}] &,
1]; \[IndentingNewLine]If[n \[Equal] {}, i = orden - 1,
gen = Join[gen, n];
i = n[\([1]\)]; \[IndentingNewLine]j =
Position[m[\([i]\)], 1, 1, 1]; \[IndentingNewLine]filas =
Select[Range[i + 1, orden - 1],
m[\([#1, j[\([1, 1]\)]]\)] \[Equal]
1 &]; \[IndentingNewLine]Do[
m = ReplacePart[m,
Mod[m[\([filas[\([k]\)]]\)] + m[\([i]\)], 2],
filas[\([k]\)]], {k,
Length[filas]}]];\[IndentingNewLine]]; \
\[IndentingNewLine]Do[
base2cociclos =
Append[base2cociclos,
Partition[
Replace[
Replace[cobordes[\([gen[\([i]\)]]\)], 1 \[Rule] \(-1\),
1], 0 \[Rule] 1, 1],
orden]]; \[IndentingNewLine]Print["\",
gen[\([i]\)] + 1, "\<-ith 2-coboundary is a generator:\>"];
Print[base2cociclos[\([i]\)] // MatrixForm], {i,
Length[gen]}]; \[IndentingNewLine] (*\ \[IndentingNewLine]\
\[IndentingNewLine]Constructing\ a\ basis\ for\ representative\ 2 -
cocycles\ coming\ from\ \(\(inflation\)\(.\)\)\
\[IndentingNewLine]\[IndentingNewLine]*) \[IndentingNewLine]Print["\<\
Calculating a system of generators for 2-cocycles coming from \
inflation...\>"]; \[IndentingNewLine]k =
Select[Range[Length[abelianizado]],
EvenQ[abelianizado[\([#1]\)]] &];
Do[col =
Select[Range[Length[M2[\([1]\)]]],
Mod[fnsa1[\([2, 2, #1, k[\([m1]\)]]\)],
abelianizado[\([k[\([m1]\)]]\)]] \[NotEqual] 0 &];
matriz = {Table[1, {i, orden}]};
m2 = 2^\(FactorInteger[abelianizado[\([k[\([m1]\)]]\)]]\)[\([1,
2]\)]; Do[matriz = Append[matriz, {1}];
Do[mi = F1[\([i]\)]; mj = F1[\([j]\)];
n = Mod[
Apply[Plus,
Table[fnsa1[\([2, 2, col[\([m3]\)], k[\([m1]\)]]\)]*
mi[\([col[\([m3]\)]]\)], {m3, Length[col]}]],
m2] + Mod[
Apply[Plus,
Table[fnsa1[\([2, 2, col[\([m3]\)], k[\([m1]\)]]\)]*
mj[\([col[\([m3]\)]]\)], {m3, Length[col]}]],
m2]; matriz[\([i]\)] =
Append[matriz[\([i]\)], \((\(-1\))\)^Floor[n/m2]], {j, 2,
orden}], {i, 2, orden}];
base2cociclos = Append[base2cociclos, matriz];
Print[matriz // MatrixForm], {m1,
Length[k]}]; \[IndentingNewLine] (*\ \[IndentingNewLine]\
\[IndentingNewLine]Constructing\ a\ basis\ for\ representative\ 2 -
cocycles\ coming\ from\ \(\(transgression\)\(.\)\)\
\[IndentingNewLine]\[IndentingNewLine]*) \[IndentingNewLine]Print["\<\
Calculating a system of generators for 2-cocycles coming from \
transgression...\>"]; \[IndentingNewLine]k =
Select[Range[Length[homologia]], EvenQ[homologia[\([#1]\)]] &];
Do[col =
Select[Range[b1], OddQ[fnsa2[\([2, 2, #1, k[\([m1]\)]]\)]] &];
matriz = {Table[1, {i, orden}]};
Do[matriz = Append[matriz, {1}];
Do[m = F2[\([\((i - 1)\)*orden + j]\)];
m2 = Mod[
Apply[Plus,
Table[m[\([col[\([m3]\)]]\)], {m3, Length[col]}]], 2];
matriz[\([i]\)] =
Append[matriz[\([i]\)], \((\(-1\))\)^m2], {j, 2,
orden}], {i, 2, orden}];
base2cociclos = Append[base2cociclos, matriz];
Print[matriz // MatrixForm], {m1,
Length[k]}]; \[IndentingNewLine] (*\[IndentingNewLine]\ \
\[IndentingNewLine]Searching\ for\ Hadamard\ cocyclic\ matrices\
\[IndentingNewLine]\[IndentingNewLine]*) \[IndentingNewLine]If[
method \[Equal]
1, \[IndentingNewLine] (*\[IndentingNewLine]\ \
\[IndentingNewLine]Exhaustive\ search\[IndentingNewLine]\[IndentingNewLine]\
*) \[IndentingNewLine]had = {}; \[IndentingNewLine]Print["\"]; \[IndentingNewLine]Do[
ele = Position[Reverse[IntegerDigits[i, 2]],
1]; \[IndentingNewLine]m =
Fold[prodhadamard, matrizdeunos[orden],
Extract[base2cociclos, ele]]; \[IndentingNewLine]If[
testhadamard[
m], \[IndentingNewLine]Print["\",
Flatten[ele,
1], "\< gives raise to a Hadamard matrix\>"]; (*\(Print[
m // MatrixForm];\)*) \[IndentingNewLine]had =
Append[had, m]]\[IndentingNewLine], {i,
2^Length[base2cociclos] -
1}]; \[IndentingNewLine]Print["\",
Length[had], "\< Hadamard matrices coming from normalized \
2-cocycles.\>"], \[IndentingNewLine] (*\[IndentingNewLine]\ \
\[IndentingNewLine]Heuristic\ \(search : \
a\ genetic\ algorithm\)\[IndentingNewLine]\
\[IndentingNewLine]*) \[IndentingNewLine]t =
orden/4; \[IndentingNewLine]RandomPermutation[
n_Integer?Positive] :=
Block[{t},
t = Array[{Random[], #} &, n]; \[IndentingNewLine]t =
Sort[t];
Map[#[\([2]\)] &,
t]]; \[IndentingNewLine]Print["\"]; \[IndentingNewLine] (*\
La\ poblaci\[OAcute]n\ constar\[AAcute]\ siempre\ de\ 4
t\ individuos, \
que\ se\ reproducir\[AAcute]n\ mediante\ cruces\ simples . \
La\ probabilidad\ de\ mutaci\[OAcute]n\ ser\[AAcute]\ del\ \
1\ por\ ciento . \
Se\ permite\ la\ existencia\ de\ \(\(gemelos\)\(.\)\)\ \
*) \[IndentingNewLine]adaptacion[p_] :=
Map[adaptacion1, p]; \[IndentingNewLine]adaptacion1[l_] :=
Module[{k, n}, k = Position[l, 1];
k = Fold[prodhadamard, matrizdeunos[4*t],
Extract[base2cociclos, k]]; n = 0;
Do[If[Apply[Plus, k[\([i]\)]] \[Equal] 0, n = n + 1], {i,
4*t}]; n]; \[IndentingNewLine]cruces[p_] :=
Module[{k, l, n}, n = 2*Floor[Length[p]/2];
l = RandomPermutation[n]; k = {};
Do[k = Join[k,
cruces1[p[\([l[\([2*i - 1]\)]]\)],
p[\([l[\([2*i]\)]]\)]]], {i, n/2}];
k]; \[IndentingNewLine]cruces1[k_, l_] :=
Module[{m, n},
n = Random[Integer, {1, Length[base2cociclos] - 1}]; {Join[
Take[k, n],
Take[l, \(-\((Length[base2cociclos] - n)\)\)]],
Join[Take[k, \(-\((Length[base2cociclos] - n)\)\)],
Take[l, n]]}]; \[IndentingNewLine]mutacion[p_] :=
Module[{n, k},
k = Table[Random[Integer, {1, 100}], {i, Length[p]}];
n = Select[Range[Length[p]], k[\([#1]\)] \[LessEqual] 10 &];
k = {};
Do[k = Append[k, mutacion1[p[\([n[\([i]\)]]\)]]], {i,
Length[n]}]; k]; \[IndentingNewLine]mutacion1[l_] :=
Module[{n, k}, n = Random[Integer, {1, Length[base2cociclos]}];
k = l; ReplacePart[k, 1 - l[\([n]\)],
n]]; \[IndentingNewLine]pob = {};
iter = 0; \[IndentingNewLine]Do[
pob = Append[pob,
Table[Random[Integer], {j, Length[base2cociclos]}]], {i,
4*t}]; \[IndentingNewLine]mathad = adaptacion[pob];
pos = Flatten[Position[mathad, 4*t - 1],
1]; \[IndentingNewLine]While[Length[pos] \[Equal] 0,
iter = iter + 1;
Print["\", iter];
descen = mutacion[pob]; pob = Join[pob, descen];
mathad = Join[mathad, adaptacion[descen]];
descen = Union[cruces[pob]];
Print["\"];
mathad = Join[mathad, adaptacion[descen]];
Print["\"];
pob = Join[pob, descen]; ind = 1; pob2 = {}; numero = 0;
mathad2 = {}; \[IndentingNewLine]While[numero \[LessEqual] 4*t,
natalidad = Flatten[Position[mathad, 4*t - ind], 1];
Print["\",
ind - 1, "\< rows to be Hadamard\>"];
config =
Union[Table[
pob[\([natalidad[\([i]\)]]\)], {i,
Length[natalidad]}]]; pob2 = Join[pob2, config];
mathad2 =
Join[mathad2, Table[4*t - ind, {i, Length[config]}]];
numero = numero + Length[config];
Print[Length[config], "\< individuals\>"];
ind = ind +
1]; \[IndentingNewLine]Print["\"]; \[IndentingNewLine]pob = pob2;
mathad = mathad2; \[IndentingNewLine]If[ind \[NotEqual] 3,
mathad = Take[mathad2, 4*t];
auxiliar =
Table[Random[
Integer, {numero - Length[config] + 1,
numero + 1 - i}], {i, numero - 4*t}];
Do[pob = Delete[pob, auxiliar[\([i]\)]], {i,
Length[auxiliar]}]]; \[IndentingNewLine]Do[
pob = Append[pob,
Table[Random[Integer], {j, Length[base2cociclos]}]], {i,
t}]; mathad = Join[mathad, adaptacion[Take[pob, \(-t\)]]];
pos = Flatten[Position[mathad, 4*t - 1],
1]]; \[IndentingNewLine]Print["\",
iter, "\< generations...\>"]; \[IndentingNewLine]Do[
Print[pob[\([pos[\([i]\)]]\)]], {i,
Length[pos]}];\[IndentingNewLine]]];\)\)\)], "Input",
InitializationCell->True]
}, Open ]],
Cell[CellGroupData[{
Cell["End the private context", "Subsection",
InitializationCell->True],
Cell[CellGroupData[{
Cell["End[ ]", "Input",
InitializationCell->True],
Cell[BoxData[
\("Global`HadamardSearch`Private`"\)], "Output"]
}, Open ]]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["Epilog", "Section",
InitializationCell->True],
Cell["This section ends the package.", "Text"],
Cell[CellGroupData[{
Cell["End the package context", "Subsection",
InitializationCell->True],
Cell["EndPackage[ ]", "Input",
InitializationCell->True]
}, Open ]]
}, Open ]],
Cell[CellGroupData[{
Cell["Examples, Tests", "Section"],
Cell[TextData[StyleBox["Examples, tests for the use of the package can go \
here.",
FontSlant->"Italic"]], "Text"],
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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{"8", "7", "6", "5", "4", "3", "2", "1", "16", "15", "14",
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{"9", "10", "11", "12", "13", "14", "15", "16", "1", "2", "3",
"4", "5", "6", "7", "8"},
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"5", "4", "3", "2", "1"}
}], "\[NoBreak]", ")"}],
(MatrixForm[ #]&)]}], ";"}]], "Input"],
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[TextData[{
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Cell[BoxData[
\(\(\(A\)\(\ \)\(matrix\)\(\ \)\(representing\)\(\ \)\(M2\)\(\ \)\((d2)\
\)\(\ \)\(is\)\(\ \)\)\)]]
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(MatrixForm[ #]&)], ";"}]], "Input"],
Cell["\<\
A matrix representing M3 (d3) is \
\>", "Text"],
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}], "\[NoBreak]", ")"}]}], ";"}]], "Input"],
Cell["\<\
Now we may proceed to develope an exahustive search (this is only \
feasible for groups of low order, up to order 28).\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
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Cell[BoxData[
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