void grassmann_embedded::unrank_INT(INT *subspace_basis, INT rk, INT verbose_level)
// subspace_basis is k x big_n
{
INT f_v = (verbose_level >= 1);
if (f_v) {
cout << "grassmann_embedded::unrank_INT" << endl;
cout << "rk=" << rk << endl;
cout << "calling G->unrank_INT" << endl;
}
G->unrank_INT(rk, verbose_level);
if (f_v) {
cout << "grassmann_embedded::unrank_INT coefficient matrix:" << endl;
print_integer_matrix_width(cout, G->M, k, n, n, F->log10_of_q);
}
if (f_v) {
cout << "grassmann_embedded::rank_INT subspace_basis:" << endl;
print_integer_matrix_width(cout, M, n, big_n, big_n, F->log10_of_q);
}
F->mult_matrix_matrix(G->M, M, subspace_basis, k, n, big_n);
if (f_v) {
cout << "grassmann_embedded::unrank_INT subspace_basis:" << endl;
print_integer_matrix_width(cout, subspace_basis, k, big_n, big_n, F->log10_of_q);
}
}
void unusual_model::convert_from_usual(INT n, INT *usual_coordinates, INT *unusual_coordinates, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT i, j, a, b, c, aa, bb;
INT *tmp;
tmp = NEW_INT(n * 4);
if (f_v) {
cout << "convert_from_usual:" << endl;
print_integer_matrix_width(cout, usual_coordinates, n, 5, 5, 2);
}
if (q == 0) {
cout << "q=" << q << " is zero" << endl;
exit(1);
}
for (i = 0; i < n; i++) {
f.mult_matrix(usual_coordinates + i * 5 + 1, hyperbolic_basis, tmp + i * 4, 1, 4, 4);
}
if (f_v) {
cout << "tmp:" << endl;
print_integer_matrix_width(cout, tmp, n, 4, 4, 2);
}
for (i = 0; i < n; i++) {
for (j = 0; j < 2; j++) {
a = tmp[i * 4 + j * 2 + 0];
b = tmp[i * 4 + j * 2 + 1];
//c = b * q + a;
c = pair_embedding[a * q + b];
aa = components[c * 2 + 0];
bb = components[c * 2 + 1];
if (aa != a) {
cout << "aa=" << aa << " not equal to a=" << a << endl;
cout << "a=" << a << " b=" << b << " c=" << c << endl;
cout << "a * q + b = " << a * q + b << endl;
cout << "q=" << q << endl;
cout << "aa=" << aa << endl;
cout << "bb=" << bb << endl;
exit(1);
}
if (bb != b) {
cout << "bb=" << bb << " not equal to b=" << b << endl;
cout << "a=" << a << " b=" << b << " c=" << c << endl;
cout << "a * q + b = " << a * q + b << endl;
cout << "aa=" << aa << endl;
cout << "bb=" << bb << endl;
exit(1);
}
unusual_coordinates[i * 3 + j] = c;
}
unusual_coordinates[i * 3 + 2] = usual_coordinates[i * 5 + 0];
}
if (f_v) {
cout << "unusual_coordinates:" << endl;
print_integer_matrix_width(cout, unusual_coordinates, n, 3, 3, 2);
}
FREE_INT(tmp);
}
INT action_on_grassmannian::compute_image_INT_embedded(action *A, INT *Elt,
INT i, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT j, h;
if (f_v) {
cout << "action_on_grassmannian::compute_image_INT_embedded i = " << i << endl;
cout << "calling GE->unrank_INT" << endl;
}
GE->unrank_INT(subspace_basis, i, verbose_level - 1);
if (f_vv) {
cout << "action_on_grassmannian::compute_image_INT_embedded subspace_basis:" << endl;
cout << "k=" << k << endl;
cout << "big_n=" << big_n << endl;
print_integer_matrix_width(cout, subspace_basis, k, big_n, big_n, F->log10_of_q);
}
for (h = 0; h < k; h++) {
A->element_image_of_low_level(
subspace_basis + h * big_n, subspace_basis2 + h * big_n, Elt, verbose_level - 1);
}
//A->element_image_of_low_level(subspace_basis, subspace_basis2, Elt, verbose_level - 1);
#if 0
F->mult_matrix_matrix(subspace_basis, Elt, subspace_basis2, k, big_n, big_n);
if (f_vv) {
cout << "action_on_grassmannian::compute_image_INT_embedded after mult_matrix_matrix:" << endl;
print_integer_matrix_width(cout, subspace_basis2, k, big_n, big_n, F->log10_of_q);
}
if (M->f_semilinear) {
f = Elt[big_n * big_n];
if (f_v) {
cout << "f_semilinear is TRUE, f=" << f << endl;
}
F->vector_frobenius_power_in_place(subspace_basis2, k * big_n, f);
}
#endif
if (f_vv) {
cout << "action_on_grassmannian::compute_image_INT_embedded subspace_basis after the action:" << endl;
print_integer_matrix_width(cout, subspace_basis2, k, big_n, big_n, F->log10_of_q);
}
j = GE->rank_INT(subspace_basis2, verbose_level - 1);
if (f_v) {
cout << "action_on_grassmannian::compute_image_INT_embedded image of " << i << " is " << j << endl;
}
return j;
}
INT action_on_subspaces::compute_image_INT(action *A, INT *Elt,
INT i, INT verbose_level)
{
INT k;
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT h, j;
if (f_v) {
cout << "action_on_subspaces::compute_image_INT_ordinary i = " << i << endl;
cout << "using action " << A->label << endl;
}
S->unrank_INT(i, verbose_level - 1);
k=S->unrank_k(i,verbose_level-1);
if (f_vv) {
cout << "action_on_subspaces::compute_image_INT_ordinary after S->unrank_INT" << endl;
print_integer_matrix_width(cout, S->G[k]->M, k, S->n, S->n, M->GFq->log10_of_q);
}
for (h = 0; h < k; h++) {
A->element_image_of_low_level(S->G[k]->M + h * n, M1 + h * n, Elt, verbose_level - 1);
}
INT_vec_copy(M1, S->G[k]->M, k * n);
j = S->rank_INT(k,verbose_level - 1);
if (f_v) {
cout << "action_on_subspaces::compute_image_INT_ordinary image of " << i << " is " << j << endl;
}
return j;
}
void unusual_model::convert_from_ranks(INT n, INT *ranks, INT *unusual_coordinates, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT *usual;
INT i;
if (f_v) {
cout << "unusual_model::convert_from_ranks" << endl;
}
if (f_v) {
cout << "ranks:" << endl;
INT_vec_print(cout, ranks, n);
cout << endl;
}
usual = NEW_INT(n * 5);
for (i = 0; i < n; i++) {
Q_unrank(f, usual + 5 * i, 1, 4, ranks[i]);
}
convert_from_usual(n, usual, unusual_coordinates, verbose_level - 1);
if (f_v) {
cout << "unusual_coordinates:" << endl;
print_integer_matrix_width(cout, unusual_coordinates, n, 3, 3, 2);
}
FREE_INT(usual);
}
void action_on_grassmannian::compute_image_longinteger(action *A, INT *Elt,
longinteger_object &i, longinteger_object &j, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT h;
if (f_v) {
cout << "action_on_grassmannian::compute_image_longinteger i = " << i << endl;
}
G->unrank_longinteger(i, 0/*verbose_level - 1*/);
if (f_vv) {
cout << "after G->unrank_longinteger" << endl;
print_integer_matrix_width(cout, G->M, G->k, G->n, G->n, F->log10_of_q);
}
for (h = 0; h < k; h++) {
A->element_image_of_low_level(G->M + h * n, M1 + h * n, Elt, verbose_level - 1);
}
//A->element_image_of_low_level(G->M, M1, Elt, verbose_level - 1);
#if 0
F->mult_matrix_matrix(G->M, Elt, M1, k, n, n);
if (M->f_semilinear) {
f = Elt[n * n];
F->vector_frobenius_power_in_place(M1, k * n, f);
}
#endif
if (f_vv) {
cout << "after element_image_of_low_level" << endl;
print_integer_matrix_width(cout, M1, G->k, G->n, G->n, F->log10_of_q);
}
INT_vec_copy(M1, G->M, k * n);
#if 0
for (h = 0; h < k * n; h++) {
G->M[h] = M1[h];
}
#endif
G->rank_longinteger(j, 0/*verbose_level - 1*/);
if (f_v) {
cout << "action_on_grassmannian::compute_image_longinteger image of " << i << " is " << j << endl;
}
}
void unusual_model::create_Linear_BLT_set(INT *BLT, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
//INT f_vv = (verbose_level >= 2);
INT i, minus_one;
INT *norm_table, nb = 0;
INT *Table;
if (f_v) {
cout << "unusual_model::create_Linear_BLT_set" << endl;
}
minus_one = F.negate(1);
norm_table = NEW_INT(qq);
for (i = 0; i < qq; i++) {
if (F.N2(i) == minus_one) {
norm_table[nb++] = i;
}
}
if (f_v) {
cout << nb << " norm -1 elements reduced:" << endl;
INT_vec_print(cout, norm_table, nb);
cout << endl;
}
if (nb != q + 1) {
cout << "nb != q + 1" << endl;
exit(1);
}
Table = NEW_INT((q + 1) * 3);
for (i = 0; i < nb; i++) {
Table[i * 3 + 0] = norm_table[i];
Table[i * 3 + 1] = 0;
Table[i * 3 + 2] = 1;
}
if (f_v) {
cout << "Table:" << endl;
print_integer_matrix_width(cout, Table, q + 1, 3, 3, 2);
}
convert_to_ranks(q + 1, Table, BLT, verbose_level);
if (f_v) {
cout << "Linear BLT set:" << endl;
INT_vec_print(cout, BLT, q + 1);
cout << endl;
}
FREE_INT(norm_table);
FREE_INT(Table);
}
void unusual_model::transform_matrix_usual_to_unusual(orthogonal *O,
INT *M5, INT *M4, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
//INT f_vvv = (verbose_level >= 3);
INT *M4_tmp1, *M4_tmp2; //, *M5;
INT i, j, a;
M4_tmp1 = NEW_INT(4 * 4);
M4_tmp2 = NEW_INT(4 * 4);
//M5 = NEW_INT(5 * 5);
if (f_v) {
cout << "unusual_model::transform_matrix_usual_to_unusual" << endl;
}
#if 0
if (f_vv) {
A->print(cout, Elt);
}
for (i = 0; i < 25; i++) {
M5[i] = Elt[i];
}
#endif
if (M5[0] != 1) {
a = f.inverse(M5[0]);
for (i = 0; i < 25; i++) {
M5[i] = f.mult(a, M5[i]);
}
}
for (i = 0; i < 4; i++) {
for (j = 0; j < 4; j++) {
a = M5[(i + 1) * 5 + j + 1];
M4_tmp2[i * 4 + j] = a;
}
}
f.mult_matrix(hyperbolic_basis_inverse, M4_tmp2, M4_tmp1, 4, 4, 4);
f.mult_matrix(M4_tmp1, hyperbolic_basis, M4, 4, 4, 4);
if (f_vv) {
cout << "transformation matrix in unusual model" << endl;
print_integer_matrix_width(cout, M4, 4, 4, 4, 3);
}
FREE_INT(M4_tmp1);
FREE_INT(M4_tmp2);
//FREE_INT(M5);
}
void unusual_model::convert_to_usual(INT n, INT *unusual_coordinates, INT *usual_coordinates, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT i, j, a, b, c;
INT *tmp;
tmp = NEW_INT(n * 4);
if (f_v) {
cout << "convert_to_usual:" << endl;
print_integer_matrix_width(cout, unusual_coordinates, n, 3, 3, 2);
}
for (i = 0; i < n; i++) {
for (j = 0; j < 2; j++) {
c = unusual_coordinates[i * 3 + j];
a = components[c * 2 + 0];
b = components[c * 2 + 1];
//a = c % q;
//b = (c - a) / q;
tmp[i * 4 + j * 2 + 0] = a;
tmp[i * 4 + j * 2 + 1] = b;
}
}
if (f_v) {
cout << "tmp:" << endl;
print_integer_matrix_width(cout, tmp, n, 4, 4, 2);
}
for (i = 0; i < n; i++) {
f.mult_matrix(tmp + i * 4, hyperbolic_basis_inverse, usual_coordinates + i * 5 + 1, 1, 4, 4);
usual_coordinates[i * 5 + 0] = unusual_coordinates[i * 3 + 2];
}
if (f_v) {
cout << "usual_coordinates:" << endl;
print_integer_matrix_width(cout, usual_coordinates, n, 5, 5, 2);
}
FREE_INT(tmp);
}
void action_on_grassmannian::init_embedding(INT big_n, INT *ambient_space, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
if (f_v) {
cout << "action_on_grassmannian::init_embedding" << endl;
cout << "big_n=" << big_n << endl;
cout << "ambient space:" << endl;
print_integer_matrix_width(cout, ambient_space, n, big_n, big_n, F->log10_of_q);
}
action_on_grassmannian::big_n = big_n;
f_embedding = TRUE;
GE = new grassmann_embedded;
GE->init(big_n, n, G, ambient_space, verbose_level);
subspace_basis = NEW_INT(n * big_n);
subspace_basis2 = NEW_INT(n * big_n);
}
INT action_on_grassmannian::compute_image_INT_ordinary(action *A, INT *Elt,
INT i, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT h, j;
if (f_v) {
cout << "action_on_grassmannian::compute_image_INT_ordinary i = " << i << endl;
cout << "A->low_level_point_size=" << A->low_level_point_size << endl;
cout << "using action " << A->label << endl;
}
G->unrank_INT(i, verbose_level - 1);
if (f_vv) {
cout << "action_on_grassmannian::compute_image_INT_ordinary after G->unrank_INT" << endl;
print_integer_matrix_width(cout, G->M, G->k, G->n, G->n, M->GFq->log10_of_q);
}
for (h = 0; h < k; h++) {
A->element_image_of_low_level(G->M + h * n, M1 + h * n, Elt, verbose_level - 1);
}
#if 0
F->mult_matrix_matrix(G->M, Elt, M1, k, n, n);
if (M->f_semilinear) {
f = Elt[n * n];
F->vector_frobenius_power_in_place(M1, k * n, f);
}
#endif
INT_vec_copy(M1, G->M, k * n);
#if 0
for (h = 0; h < k * n; h++) {
G->M[h] = M1[h];
}
#endif
j = G->rank_INT(verbose_level - 1);
if (f_v) {
cout << "action_on_grassmannian::compute_image_INT_ordinary image of " << i << " is " << j << endl;
}
return j;
}
void action_by_representation::compute_image_INT_low_level(
action &A, INT *Elt, INT *input, INT *output, INT verbose_level)
{
INT *x = input;
INT *xA = output;
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT i, f;
if (f_v) {
cout << "action_by_representation::compute_image_INT_low_level" << endl;
}
if (f_vv) {
cout << "action_by_representation::compute_image_INT_low_level: x=";
INT_vec_print(cout, x, dimension);
cout << endl;
}
INT a, b, c, d;
a = Elt[0];
b = Elt[1];
c = Elt[2];
d = Elt[3];
INT AA[9];
INT two;
two = F->add(1, 1);
AA[0] = F->mult(a, a);
AA[2] = F->mult(b, b);
AA[6] = F->mult(c, c);
AA[8] = F->mult(d, d);
AA[1] = F->mult(a, b);
AA[7] = F->mult(c, d);
AA[3] = F->product3(two, a, c);
AA[5] = F->product3(two, b, d);
AA[4] = F->add(F->mult(a, d), F->mult(b, c));
if (f_v) {
cout << "A=" << endl;
print_integer_matrix_width(cout, AA, 3, 3, 3, F->log10_of_q);
}
F->mult_matrix(x, AA, xA, 1, 3, 3);
if (f_vv) {
cout << "action_by_representation::compute_image_INT_low_level: xA=";
INT_vec_print(cout, xA, dimension);
cout << endl;
}
if (M->f_semilinear) {
f = Elt[n * n];
for (i = 0; i < dimension; i++) {
xA[i] = F->frobenius_power(xA[i], f);
}
if (f_vv) {
cout << "after " << f << " field automorphisms: xA=";
INT_vec_print(cout, xA, dimension);
cout << endl;
}
}
if (f_v) {
cout << "action_by_representation::compute_image_INT_low_level done" << endl;
}
}
void conic(INT q, INT *six_coeffs, INT xmax, INT ymax, INT f_do_stabilizer, INT verbose_level)
{
const BYTE *override_poly = NULL;
finite_field F;
projective_space *P;
action *A;
INT n = 3;
//INT f_with_group = TRUE;
INT v[3];
//INT w[3];
//F.init(q), verbose_level - 2);
F.init_override_polynomial(q, override_poly, verbose_level);
P = new projective_space;
cout << "before P->init" << endl;
P->init(n - 1, &F,
FALSE /* f_init_incidence_structure */,
verbose_level/*MINIMUM(2, verbose_level)*/);
cout << "after P->init" << endl;
A = new action;
A->init_general_linear_group(n, &F, FALSE /* f_semilinear */, TRUE /* f_basis */, verbose_level - 2);
INT variety[100];
INT variety_size = 0;
INT a, i, j, h;
for (i = 0; i < P->N_points; i++) {
P->unrank_point(v, i);
a = F.evaluate_conic_form(six_coeffs, v);
cout << i << " : ";
INT_vec_print(cout, v, 3);
cout << " : " << a << endl;
if (a == 0) {
variety[variety_size++] = i;
}
}
cout << "the size of the variety is " << variety_size << endl;
cout << "the variety is:" << endl;
INT_vec_print(cout, variety, variety_size);
cout << endl;
for (i = 0; i < variety_size; i++) {
P->unrank_point(v, variety[i]);
a = F.evaluate_conic_form(six_coeffs, v);
cout << i << " : ";
INT_vec_print(cout, v, 3);
cout << " : " << a << endl;
}
#if 0
INT pts[] = {2, 22, 6, 18, 10};
find_collinear_triple(P, pts, 5);
INT five_pts[] = {3, 6, 7, 28, 30};
#endif
//INT six_coeffs[6];
INT five_pts[5];
five_pts[0] = variety[0];
five_pts[1] = variety[1];
five_pts[2] = variety[2];
five_pts[3] = variety[3];
five_pts[4] = variety[4];
P->determine_conic_in_plane(five_pts, 5, six_coeffs, verbose_level);
INT points[1000];
INT tangents[1000];
INT *exterior_points;
INT *secants;
INT nb_secants, nb_exterior_points;
INT nb_points;
//INT v[3];
P->conic_points(five_pts, six_coeffs, points, nb_points, verbose_level);
cout << "the " << nb_points << " conic points are: ";
INT_vec_print(cout, points, nb_points);
cout << endl;
for (i = 0; i < nb_points; i++) {
P->unrank_point(v, points[i]);
cout << i << " : " << points[i] << " : ";
INT_vec_print(cout, v, 3);
cout << endl;
}
strong_generators *Aut_gens;
if (f_do_stabilizer) {
// computing stabilizer:
cout << "computing stabilizer" << endl;
set_stabilizer_compute STAB;
sims *Stab;
INT nb_backtrack_nodes;
cout << "computing stabilizer of conic:" << endl;
STAB.init(A, variety, variety_size /* points, nb_points*/ , verbose_level);
STAB.compute_set_stabilizer(t0, nb_backtrack_nodes, Aut_gens, verbose_level + 10);
longinteger_object go, go2;
Stab = Aut_gens->create_sims(verbose_level - 1);
Stab->group_order(go);
cout << "computing stabilizer of conic done, found a group of order " << go << endl;
delete Stab;
}
else {
Aut_gens = NULL;
}
{
BYTE fname[1000];
{
mp_graphics *G;
INT f_include_line_at_infinity = TRUE;
sprintf(fname, "conic_%ld", q);
draw_beginning(fname, G, xmax, ymax, verbose_level);
//variety_size = 0;
draw_grid_(*G, q, f_include_line_at_infinity, verbose_level);
draw_points(*G, P, /*variety + 3, 5*/ variety, variety_size /*points, nb_points*/, verbose_level);
draw_end(fname, G, xmax, ymax, verbose_level);
}
if (f_do_stabilizer) {
prepare_latex(fname, P, points, nb_points, Aut_gens, verbose_level);
}
}
delete Aut_gens;
return;
P->find_tangent_lines_to_conic(six_coeffs,
points, nb_points,
tangents, verbose_level);
cout << "the " << nb_points << " tangent lines are: ";
INT_vec_print(cout, tangents, nb_points);
cout << endl;
nb_exterior_points = (nb_points * (nb_points - 1)) >> 1;
nb_secants = nb_exterior_points;
exterior_points = NEW_INT(nb_exterior_points);
h = 0;
for (i = 0; i < nb_points; i++) {
for (j = i + 1; j < nb_points; j++) {
exterior_points[h++] = P->Line_intersection[
tangents[i] * P->N_lines + tangents[j]];
}
}
INT_vec_heapsort(exterior_points, nb_exterior_points);
cout << "the " << nb_exterior_points << " exterior points are: ";
INT_vec_print(cout, exterior_points, nb_exterior_points);
cout << endl;
secants = NEW_INT(nb_secants);
h = 0;
for (i = 0; i < nb_points; i++) {
for (j = i + 1; j < nb_points; j++) {
secants[h++] = P->Line_through_two_points[
points[i] * P->N_points + points[j]];
}
}
INT_vec_heapsort(secants, nb_secants);
cout << "the " << nb_secants << " secants are: ";
INT_vec_print(cout, secants, nb_secants);
cout << endl;
INT *external_lines;
INT nb_external_lines;
external_lines = NEW_INT(P->N_lines);
for (i = 0; i < P->N_lines; i++) {
external_lines[i] = i;
}
nb_external_lines = P->N_lines;
set_delete_elements(external_lines, nb_external_lines, tangents, nb_points);
set_delete_elements(external_lines, nb_external_lines, secants, nb_secants);
cout << "the " << nb_external_lines << " external lines are: ";
INT_vec_print(cout, external_lines, nb_external_lines);
cout << endl;
INT *adjacency;
INT idx;
adjacency = NEW_INT(nb_external_lines * nb_external_lines);
for (i = 0; i < nb_external_lines; i++) {
adjacency[i * nb_external_lines + i] = 0;
for (j = i + 1; j < nb_external_lines; j++) {
a = P->Line_intersection[
external_lines[i] * P->N_lines + external_lines[j]];
if (INT_vec_search(exterior_points, nb_exterior_points, a, idx)) {
adjacency[i * nb_external_lines + j] = 1;
adjacency[j * nb_external_lines + i] = 1;
}
else {
adjacency[i * nb_external_lines + j] = 0;
adjacency[j * nb_external_lines + i] = 0;
}
}
}
cout << "adjacency matrix:" << endl;
print_integer_matrix_width(cout, adjacency,
nb_external_lines, nb_external_lines, nb_external_lines,
1);
INT *Edges1;
INT *Edges2;
INT *Incidence;
INT nb_e = 0;
for (i = 0; i < nb_external_lines; i++) {
for (j = i + 1; j < nb_external_lines; j++) {
if (adjacency[i * nb_external_lines + j])
nb_e++;
}
}
Edges1 = NEW_INT(nb_e);
Edges2 = NEW_INT(nb_e);
Incidence = NEW_INT(nb_external_lines * nb_e);
for (i = 0; i < nb_external_lines * nb_e; i++) {
Incidence[i] = 0;
}
nb_e = 0;
for (i = 0; i < nb_external_lines; i++) {
for (j = i + 1; j < nb_external_lines; j++) {
if (adjacency[i * nb_external_lines + j]) {
Edges1[nb_e] = i;
Edges2[nb_e] = j;
nb_e ++;
}
}
}
for (j = 0; j < nb_e; j++) {
Incidence[Edges1[j] * nb_e + j] = 1;
Incidence[Edges2[j] * nb_e + j] = 1;
}
BYTE fname[1000];
sprintf(fname, "ext_lines_%ld.inc", P->F->q);
{
ofstream f(fname);
f << nb_external_lines << " " << nb_e << " " << nb_e * 2 << endl;
for (i = 0; i < nb_external_lines * nb_e; i++) {
if (Incidence[i]) {
f << i << " ";
}
}
f << endl;
f << -1 << endl;
}
cout << "written file " << fname << " of size " << file_size(fname) << endl;
INT colors[] = {0,1};
INT C = 2;
INT *Pijk;
Pijk = NEW_INT(C * C * C);
if (analyze_color_graph(C, colors, nb_external_lines, adjacency, Pijk, verbose_level)) {
cout << "is association scheme" << endl;
}
else {
cout << "not an association scheme" << endl;
}
//exit(1);
{
set_stabilizer_compute STAB;
strong_generators *Aut_gens;
sims *Stab;
//vector_ge gens;
//INT *tl;
INT nb_backtrack_nodes;
cout << "computing stabilizer of conic:" << endl;
STAB.init(A, points, nb_points, verbose_level);
STAB.compute_set_stabilizer(t0, nb_backtrack_nodes, Aut_gens, verbose_level + - 2);
Stab = Aut_gens->create_sims(verbose_level - 1);
longinteger_object go, go2;
Stab->group_order(go);
cout << "computing stabilizer of conic done, found a group of order " << go << endl;
action *A2;
action A2r;
INT f_induce_action = TRUE;
projective_space_init_line_action(P, A, A2, verbose_level);
A2r.induced_action_by_restriction(*A2,
f_induce_action, Stab,
nb_external_lines, external_lines, verbose_level);
A2r.group_order(go2);
cout << "induced action on external lines has order " << go2 << endl;
for (i = 0; i < Aut_gens->gens->len; i++) {
A2r.element_print_quick(Aut_gens->gens->ith(i), cout);
A2->element_print_as_permutation(Aut_gens->gens->ith(i), cout);
A2r.element_print_as_permutation(Aut_gens->gens->ith(i), cout);
}
delete Aut_gens;
delete Stab;
delete A2;
}
}
INT unusual_model::build_candidate_set_with_or_without_test(orthogonal &O, INT q,
INT gamma, INT delta, INT offset, INT m, INT *Set,
INT f_second_half, INT f_test, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT f_vvv = (verbose_level >= 3);
INT i, z2i, z2mi;
INT len = (q + 1) / 2;
INT Len = 0;
INT *Table;
INT zeta;
Table = NEW_INT((q + 1) * 3);
zeta = F.alpha_power(q - 1);
for (i = 0; i < len; i++) {
z2i = F.power(zeta, 2 * i);
z2mi = F.power(z2i, m);
Table[i * 3 + 0] = F.mult(gamma, z2i);
Table[i * 3 + 1] = F.mult(delta, z2mi);
Table[i * 3 + 2] = 1;
}
Len += len;
convert_to_ranks(Len, Table, Set, verbose_level - 2);
if (f_vvv) {
cout << "created the following 1st half:" << endl;
INT_vec_print(cout, Set, Len);
cout << endl;
print_coordinates_detailed_set(Set, Len);
}
if (f_test) {
for (i = 1; i < Len; i++) {
if (!O.BLT_test_full(i, Set, 0/*verbose_level*/)) {
cout << "BLT test fails in point " << i << " in 1st half" << endl;
FREE_INT(Table);
return FALSE;
}
}
if (f_vv) {
cout << "passes BLT test for 1st half" << endl;
}
}
if (f_second_half) {
INT z2s;
for (i = 0; i < len; i++) {
z2i = F.power(zeta, 2 * i);
z2mi = F.power(z2i, m);
z2s = F.power(zeta, 2 * offset);
Table[(len + i) * 3 + 0] = F.mult(delta, z2i);
Table[(len + i) * 3 + 1] = F.mult(F.mult(gamma, z2mi), z2s);
Table[(len + i) * 3 + 2] = 1;
}
Len += len;
convert_to_ranks(Len, Table, Set, verbose_level - 2);
if (f_test) {
for (i = 1; i < len; i++) {
if (!O.BLT_test_full(i, Set + len, 0/*verbose_level*/)) {
cout << "BLT test fails in point " << i << " in 2nd half" << endl;
FREE_INT(Table);
return FALSE;
}
}
if (f_vv) {
cout << "passes BLT test for second half" << endl;
}
}
}
if (FALSE) {
cout << "Table:" << endl;
print_integer_matrix_width(cout, Table, Len, 3, 3, 2);
}
convert_to_ranks(Len, Table, Set, verbose_level - 2);
if (f_vvv) {
cout << "created the following set:" << endl;
INT_vec_print(cout, Set, Len);
cout << endl;
print_coordinates_detailed_set(Set, Len);
}
#if 0
//INT_vec_sort(Len, Set);
for (i = 0; i < Len - 1; i++) {
if (Set[i] == Set[i + 1]) {
cout << "the set contains repeats" << endl;
FREE_INT(Table);
return FALSE;
}
}
#endif
if (f_test) {
for (i = 1; i < Len; i++) {
if (!O.BLT_test(i, Set, 0/*verbose_level*/)) {
if (f_v) {
cout << "BLT test fails in point " << i << " in the joining" << endl;
}
FREE_INT(Table);
return FALSE;
}
}
if (f_v) {
cout << "passes BLT test" << endl;
}
}
if (Len < q + 1) {
FREE_INT(Table);
return FALSE;
}
FREE_INT(Table);
return TRUE;
}
void unusual_model::create_Mondello_BLT_set(INT *BLT, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
//INT f_vv = (verbose_level >= 2);
INT i, beta, gamma;
INT *norm_one_table, nb_norm_one = 0;
INT *Table;
INT minus_one, four, five, minus_four_fifth, minus_one_fifth;
if (f_v) {
cout << "unusual_model::create_Mondello_BLT_set" << endl;
}
minus_one = F.negate(1);
four = 4 % F.p;
five = 5 % F.p;
minus_four_fifth = F.negate(F.mult(four, F.inverse(five)));
minus_one_fifth = F.negate(F.inverse(five));
// now we find an element beta in F_q^2 with N2(beta) = minus_four_fifth
for (beta = 1; beta < qq; beta++) {
if (F.N2(beta) == minus_four_fifth) {
break;
}
}
if (beta == qq) {
cout << "did not find beta" << endl;
}
if (f_v) {
cout << "beta=" << beta << endl;
}
// now we find an element gamma in F_q^2 with N2(beta) = minus_one_fifth
for (gamma = 1; gamma < qq; gamma++) {
if (F.N2(gamma) == minus_one_fifth) {
break;
}
}
if (gamma == qq) {
cout << "did not find gamma" << endl;
}
if (f_v) {
cout << "gamma=" << gamma << endl;
}
norm_one_table = NEW_INT(qq);
for (i = 0; i < qq; i++) {
if (F.N2(i) == 1) {
norm_one_table[nb_norm_one++] = i;
}
}
if (f_v) {
cout << nb_norm_one << " norm one elements:" << endl;
INT_vec_print(cout, norm_one_table, nb_norm_one);
cout << endl;
}
if (nb_norm_one != q + 1) {
cout << "nb_norm_one != q + 1" << endl;
exit(1);
}
Table = NEW_INT((q + 1) * 3);
for (i = 0; i < q + 1; i++) {
Table[i * 3 + 0] = F.mult(beta, F.power(norm_one_table[i], 2));
Table[i * 3 + 1] = F.mult(gamma, F.power(norm_one_table[i], 3));
Table[i * 3 + 2] = 1;
}
if (f_v) {
cout << "Table:" << endl;
print_integer_matrix_width(cout, Table, q + 1, 3, 3, 2);
}
convert_to_ranks(q + 1, Table, BLT, verbose_level);
if (f_v) {
cout << "Mondello BLT set:" << endl;
INT_vec_print(cout, BLT, q + 1);
cout << endl;
}
FREE_INT(norm_one_table);
FREE_INT(Table);
}
void coding_theory_domain::create_matrix_H_subfield(
finite_field *F, finite_field*f,
int *H_subfield, int *C, int *C_inv, int *M,
int m, int n, int beta, int beta_q,
int f_elements_exponential, const char *symbol_for_print,
const char *symbol_for_print_subfield,
int f_construction_A, int f_hyperoval, int f_construction_B,
int verbose_level)
{
int f_v = (verbose_level >= 1);
int f_vv = (verbose_level >= 2);
int i, j;
int q;
//int *C;
//int *C_inv;
int *H;
int *AA;
q = f->q;
// matrix C is zero:
H = NEW_int(m * n);
AA = NEW_int(m * m);
for (i = 0; i < m; i++) {
for (j = 0; j < m; j++) {
C[i * m + j] = 0;
}
}
if (f_construction_A) {
int nb_C_coeffs = 15;
int k, aa;
int C_coeffs[] = {
0, 0, 1,
1, 1, 1,
2, 2, 1,
3, 3, 1,
3, 4, 1,
4, 3, beta_q,
4, 4, beta,
5, 5, 1,
5, 6, 1,
6, 5, beta_q,
6, 6, beta,
7, 7, 1,
7, 8, 1,
8, 7, beta_q,
8, 8, beta,
};
for (k = 0; k < nb_C_coeffs; k++) {
i = C_coeffs[k * 3 + 0];
j = C_coeffs[k * 3 + 1];
aa = C_coeffs[k * 3 + 2];
C[i * m + j] = aa;
}
}
else if (f_construction_B) {
int nb_C_coeffs = 20;
int k, aa;
int C_coeffs[] = {
0, 0, 1,
1, 1, 1,
2, 2, 1,
2, 3, 1,
2, 4, 1,
3, 2, beta,
3, 3, beta_q,
3, 4, F->beta_trinomial(q, beta, 1, 0, 0),
4, 2, F->beta_trinomial(q, beta, 0, 0, 2),
4, 3, F->beta_trinomial(q, beta, 0, 2, 0),
4, 4, F->beta_trinomial(q, beta, 2, 0, 0),
5, 5, 1,
5, 6, 1,
5, 7, 1,
6, 5, F->beta_trinomial(q, beta, 0, 1, 1),
6, 6, F->beta_trinomial(q, beta, 1, 1, 0),
6, 7, F->beta_trinomial(q, beta, 1, 0, 1),
7, 5, F->beta_trinomial(q, beta, 0, 2, 2),
7, 6, F->beta_trinomial(q, beta, 2, 2, 0),
7, 7, F->beta_trinomial(q, beta, 2, 0, 2),
};
for (k = 0; k < nb_C_coeffs; k++) {
i = C_coeffs[k * 3 + 0];
j = C_coeffs[k * 3 + 1];
aa = C_coeffs[k * 3 + 2];
C[i * m + j] = aa;
}
}
if (f_v) {
cout << "matrix C:" << endl;
print_integer_matrix_width(cout, C, m, m, m, 2);
F->latex_matrix(cout, f_elements_exponential,
symbol_for_print, C, m, m);
}
F->invert_matrix(C, C_inv, m);
if (f_vv) {
cout << "C_inv:" << endl;
print_integer_matrix_width(cout, C_inv, m, m, m, 2);
}
F->mult_matrix_matrix(C, C_inv, AA, m, m, m,
0 /* verbose_level */);
if (f_vv) {
cout << "C * C_inv:" << endl;
print_integer_matrix_width(cout, AA, m, m, m, 2);
}
F->mult_matrix_matrix(C, M, H, m, m, n,
0 /* verbose_level */);
if (f_v) {
cout << "H = C * M:" << endl;
print_integer_matrix_width(cout, H, m, n, n, 2);
F->latex_matrix(cout, f_elements_exponential,
symbol_for_print, H, m, n);
}
#if 0
rk = F.Gauss_int(M, FALSE /* f_special */, TRUE /* f_complete */, base_cols,
FALSE /* f_P */, NULL, m /* m */, n /* n */, 0 /* Pn */,
verbose_level - 2);
cout << "has rank " << rk << endl;
#endif
tt_field_reduction(*F, *f, m, n, H, H_subfield, verbose_level - 2);
if (f_v) {
cout << "H_subfield:" << endl;
print_integer_matrix_width(cout, H_subfield, m, n, n, 2);
f->latex_matrix(cout, f_elements_exponential,
symbol_for_print_subfield, H_subfield, m, n);
}
FREE_int(H);
FREE_int(AA);
}
void unusual_model::create_Fisher_BLT_set(INT *Fisher_BLT, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
//INT f_vv = (verbose_level >= 2);
INT i, j, beta, minus_one, k;
INT *norm_one_table, nb_norm_one = 0;
INT *Table;
if (f_v) {
cout << "unusual_model::create_Fisher_BLT_set" << endl;
}
minus_one = F.negate(1);
// now we find an element beta in F_q^2 with N2(beta) = -1
for (beta = 1; beta < qq; beta++) {
if (F.N2(beta) == minus_one) {
break;
}
}
if (beta == qq) {
cout << "did not find beta" << endl;
}
if (f_v) {
cout << "beta=" << beta << endl;
}
norm_one_table = NEW_INT(qq);
for (i = 0; i < qq; i++) {
if (F.N2(i) == 1) {
j = F.negate(i);
for (k = 0; k < nb_norm_one; k++) {
if (norm_one_table[k] == j)
break;
}
if (k == nb_norm_one) {
norm_one_table[nb_norm_one++] = i;
}
}
}
if (f_v) {
cout << nb_norm_one << " norm one elements reduced:" << endl;
INT_vec_print(cout, norm_one_table, nb_norm_one);
cout << endl;
}
if (nb_norm_one != (q + 1) / 2) {
cout << "nb_norm_one != (q + 1) / 2" << endl;
exit(1);
}
Table = NEW_INT((q + 1) * 3);
for (i = 0; i < nb_norm_one; i++) {
Table[i * 3 + 0] = F.mult(beta, F.mult(norm_one_table[i], norm_one_table[i]));
Table[i * 3 + 1] = 0;
Table[i * 3 + 2] = 1;
}
for (i = 0; i < nb_norm_one; i++) {
Table[(nb_norm_one + i) * 3 + 0] = 0;
Table[(nb_norm_one + i) * 3 + 1] = F.mult(beta, F.mult(norm_one_table[i], norm_one_table[i]));
Table[(nb_norm_one + i) * 3 + 2] = 1;
}
if (f_v) {
cout << "Table:" << endl;
print_integer_matrix_width(cout, Table, q + 1, 3, 3, 2);
}
convert_to_ranks(q + 1, Table, Fisher_BLT, verbose_level);
if (f_v) {
cout << "Fisher BLT set:" << endl;
INT_vec_print(cout, Fisher_BLT, q + 1);
cout << endl;
}
FREE_INT(norm_one_table);
FREE_INT(Table);
}
void W3q::init(finite_field *F, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT f_vvv = (verbose_level >= 3);
INT h, c, rk;
W3q::F = F;
W3q::q = F->q;
if (f_v) {
cout << "W3q::init" << endl;
}
P3 = new projective_space;
Q4 = new orthogonal;
Basis = NEW_INT(2 * 4);
P3->init(3, F,
//TRUE /* f_init_group */,
//FALSE /* f_line_action */,
FALSE /* f_init_incidence_structure */,
//TRUE /* f_semilinear */,
//TRUE /* f_basis */,
verbose_level - 1 /*MINIMUM(verbose_level - 1, 3)*/);
F = P3->F;
Q4->init(0, 5, F, verbose_level - 1);
Lines = NEW_INT(P3->N_lines);
nb_lines = 0;
for (h = 0; h < P3->N_lines; h++) {
P3->unrank_line(Basis, h);
c = evaluate_symplectic_form(Basis, Basis + 4);
if (c) {
continue;
}
Lines[nb_lines++] = h;
}
cout << "We found " << nb_lines << " Lines, they are" << endl;
INT_vec_print(cout, Lines, nb_lines);
cout << endl;
if (nb_lines != Q4->nb_points) {
cout << "nb_lines != Q4->nb_points" << endl;
exit(1);
}
Q4_rk = NEW_INT(nb_lines);
Line_idx = NEW_INT(nb_lines);
for (h = 0; h < nb_lines; h++) {
P3->unrank_line(Basis, Lines[h]);
if (f_vv) {
cout << "Line " << h << " is " << Lines[h] << ":" << endl;
print_integer_matrix_width(cout, Basis, 2, 4, 4, F->log10_of_q);
cout << endl;
}
isomorphism_Q4q(Basis, Basis + 4, v5);
if (f_vvv) {
cout << "v5=";
INT_vec_print(cout, v5, 5);
cout << endl;
}
rk = Q4->rank_point(v5, 1, 0);
if (f_vvv) {
cout << "orthogonal point rank " << rk << endl;
}
Q4_rk[h] = rk;
Line_idx[rk] = h;
}
if (f_v) {
cout << "The isomorphism is:" << endl;
cout << "h : Lines[h] : Q4_rk[h] : Line_idx[h] : x : y : point in Q(4,q)" << endl;
cout << "Where x and y are a basis for the line" << endl;
for (h = 0; h < nb_lines; h++) {
cout << setw(4) << h << " : ";
cout << setw(4) << Lines[h] << " : ";
cout << setw(4) << Q4_rk[h] << " : ";
cout << setw(4) << Line_idx[h] << " : ";
P3->unrank_line(Basis, Lines[h]);
INT_vec_print(cout, Basis, 4);
cout << " : ";
INT_vec_print(cout, Basis + 4, 4);
Q4->unrank_point(v5, 1, Q4_rk[h], 0);
cout << " : ";
INT_vec_print(cout, v5, 5);
cout << endl;
}
}
}
void coding_theory_domain::make_tensor_code_9_dimensional(int q,
const char *override_poly_Q, const char *override_poly,
int f_hyperoval,
int *&code, int &length,
int verbose_level)
{
finite_field F;
finite_field f;
rank_checker rc;
int exponents[9];
int *M;
int *C;
int *C_inv;
int *H;
int *H_subfield;
int index, Q, i, j, t, m, n, r, beta, beta_q;
int f_v = (verbose_level >= 1);
int f_vv = (verbose_level >= 2);
if (f_v) {
cout << "make_tensor_code_9_dimensional q=" << q << endl;
}
//q = 2; override_poly_Q = ""; override_poly = ""; int f_hyperoval = FALSE;
//q = 3; override_poly_Q = ""; override_poly = ""; int f_hyperoval = FALSE;
//q = 4; override_poly_Q = "19"; override_poly = "7"; int f_hyperoval = FALSE;
// F_256 generated by X^8 + X^4 + X^3 + X^2 + 1
// F_16 generated by X^4+X+1 = 19
// F_4 generated by X^2+X+1 = 7
//q = 5; override_poly_Q = "47"; override_poly = ""; int f_hyperoval = FALSE;
// F_625 generated by X^4 + X^3 + 3X + 2
// F_25 generated by X^2 + 4X + 2 = 47
//q = 7; override_poly_Q = ""; override_poly = ""; int f_hyperoval = FALSE;
//q = 8; override_poly_Q = "97"; override_poly = "11"; int f_hyperoval = TRUE;
// F_4096 generated by x^12+x^6+x^4+x+1
// F_64 generated by X^6+X^5+1 = 97
// F_8 generated by X^3+X+1 = 11
//q = 9; override_poly_Q = ""; override_poly = "17"; int f_hyperoval = FALSE;
beta = q;
Q = q * q;
m = 9;
if (f_hyperoval) {
n = Q + 2;
}
else {
n = Q + 1;
}
r = 5;
if (q == 4)
r = 7;
if (q == 3)
r = 9;
exponents[0] = 0;
exponents[1] = q + 1;
exponents[2] = 2 * q + 2;
exponents[3] = q;
exponents[4] = 1;
exponents[5] = 2 * q;
exponents[6] = 2;
exponents[7] = 2 * q + 1;
exponents[8] = q + 2;
//exponents[0] = 0;
//exponents[1] = q;
//exponents[2] = 2 * q;
//exponents[3] = 1;
//exponents[4] = q + 1;
//exponents[5] = 2 * q + 1;
//exponents[6] = 2;
//exponents[7] = q + 2;
//exponents[8] = 2 * q + 2;
index = (Q - 1) / (q - 1);
cout << "q = " << q << " override polynomial = " << override_poly << endl;
cout << "Q = " << Q << endl;
F.init_override_polynomial(Q, override_poly_Q, verbose_level);
cout << "field of order " << Q << " initialized" << endl;
beta_q = F.power(beta, q);
f.init_override_polynomial(q, override_poly, verbose_level);
cout << "field of order " << q << " initialized" << endl;
cout << "n = " << n << endl;
cout << "index = " << index << endl;
cout << "beta = " << beta << endl;
cout << "beta_q = " << beta_q << endl;
F.compute_subfields(verbose_level - 3);
M = NEW_int(m * n);
C = NEW_int(m * m);
C_inv = NEW_int(m * m);
H = NEW_int(m * n);
H_subfield = NEW_int(m * n);
rc.init(&f, m, n, r + 1);
for (i = 0; i < m; i++) {
for (j = 0; j < n; j++) {
M[i * n + j] = 0;
}
}
for (i = 0; i < m; i++) {
for (j = 0; j < m; j++) {
C[i * m + j] = 0;
}
}
for (t = 0; t < Q; t++) {
for (i = 0; i < m; i++) {
M[i * n + t] = F.power(t, exponents[i]);
}
}
{
M[2 * n + Q] = 1;
if (f_hyperoval)
M[1 * n + Q + 1] = 1;
int nb_C_coeffs = 15;
int k, aa;
int C_coeffs[] = {
0, 0, 1,
1, 1, 1,
2, 2, 1,
3, 3, 1,
3, 4, 1,
4, 3, beta_q,
4, 4, beta,
5, 5, 1,
5, 6, 1,
6, 5, beta_q,
6, 6, beta,
7, 7, 1,
7, 8, 1,
8, 7, beta_q,
8, 8, beta,
};
for (k = 0; k < nb_C_coeffs; k++) {
i = C_coeffs[k * 3 + 0];
j = C_coeffs[k * 3 + 1];
aa = C_coeffs[k * 3 + 2];
C[i * m + j] = aa;
}
}
cout << "M:" << endl;
print_integer_matrix_width(cout, M, m, n, n, 2);
{
int *all_one, *col_sum;
all_one = NEW_int(n);
col_sum = NEW_int(m);
for (i = 0; i < n; i++)
all_one[i] = 1;
F.mult_matrix_matrix(M, all_one, col_sum, m, n, 1,
0 /* verbose_level */);
cout << "col_sum:" << endl;
print_integer_matrix_width(cout, col_sum, m, 1, 1, 2);
FREE_int(all_one);
FREE_int(col_sum);
}
#if 0
for (j = 0; j < n; j++) {
PG_element_normalize(F, M + j, n, m);
}
cout << "column normalized M:" << endl;
print_integer_matrix_width(cout, M, m, n, n, 2);
#endif
cout << "C:" << endl;
print_integer_matrix_width(cout, C, m, m, m, 2);
F.invert_matrix(C, C_inv, m);
cout << "C_inv:" << endl;
print_integer_matrix_width(cout, C_inv, m, m, m, 2);
{
int *AA;
AA = NEW_int(m * m);
F.mult_matrix_matrix(C, C_inv, AA, m, m, m,
0 /* verbose_level */);
cout << "C * C_inv:" << endl;
print_integer_matrix_width(cout, AA, m, m, m, 2);
FREE_int(AA);
}
F.mult_matrix_matrix(C, M, H, m, m, n,
0 /* verbose_level */);
cout << "H = C * M:" << endl;
print_integer_matrix_width(cout, H, m, n, n, 2);
#if 0
rk = F.Gauss_int(M, FALSE /* f_special */, TRUE /* f_complete */, base_cols,
FALSE /* f_P */, NULL, m /* m */, n /* n */, 0 /* Pn */,
FALSE, FALSE);
cout << "has rank " << rk << endl;
#endif
if (f_vv) {
cout << "before field reduction:" << endl;
print_integer_matrix_width(cout, H, m, n, n, 2);
cout << endl;
f.print_integer_matrix_zech(cout, H, m, n);
cout << endl;
}
F.retract_int_vec(f, 2, H, H_subfield, m * n, 0 /* verbose_level */);
//field_reduction(F, f, m, n, H, H_subfield, TRUE, TRUE);
if (f_vv) {
cout << "after field reduction:" << endl;
print_integer_matrix_width(cout, H_subfield, m, n, n, 2);
cout << endl;
f.print_integer_matrix_zech(cout, H_subfield, m, n);
cout << endl;
}
cout << "H_subfield:" << endl;
print_integer_matrix_width(cout, H_subfield, m, n, n, 2);
code = H_subfield;
length = n;
FREE_int(M);
FREE_int(C);
FREE_int(C_inv);
FREE_int(H);
}
void coding_theory_domain::twisted_tensor_product_codes(
int *&H_subfield, int &m, int &n,
finite_field *F, finite_field *f,
int f_construction_A, int f_hyperoval,
int f_construction_B, int verbose_level)
{
int f_v = (verbose_level >= 1);
int f_vv = (verbose_level >= 2);
int index;
int exponents[9];
int *M;
//int *H_subfield;
int *C;
int *C_inv;
int q = f->q;
int q2;
int Q;
//int m, n;
int r;
int beta, beta_q;
int f_elements_exponential = TRUE;
const char *symbol_for_print = "\\alpha";
const char *symbol_for_print_subfield = "\\omega";
if (f_v) {
cout << "twisted_tensor_product_codes" << endl;
cout << "f_construction_A=" << f_construction_A << endl;
cout << "f_hyperoval=" << f_hyperoval << endl;
cout << "f_construction_B=" << f_construction_B << endl;
}
q2 = q * q;
Q = 0;
if (f_construction_A) {
Q = q2;
}
else if (f_construction_B) {
Q = q2 * q;
}
index = (Q - 1) / (q - 1);
if (Q != F->q) {
cout << "twisted_tensor_product_codes Q != F->q" << endl;
exit(1);
}
if (f_vv) {
cout << "q = " << q << endl;
cout << "Q = " << Q << endl;
cout << "index = " << index << endl;
}
#if 0
F.init_override_polynomial(Q, override_poly_Q, verbose_level - 2);
if (f_vv) {
cout << "field of order " << Q << " initialized" << endl;
}
f.init_override_polynomial(q, override_poly_q, verbose_level - 2);
if (f_vv) {
cout << "field of order " << q << " initialized" << endl;
cout << "index = " << index << endl;
}
#endif
F->compute_subfields(verbose_level - 2);
create_matrix_M(
M,
F, f,
m, n, beta, r, exponents,
f_construction_A, f_hyperoval, f_construction_B,
f_elements_exponential, symbol_for_print,
verbose_level - 2);
beta_q = F->power(beta, q);
if (f_vv) {
cout << "twisted_tensor_product_codes after create_matrix_M" << endl;
cout << "m = " << m << endl;
cout << "n = " << n << endl;
cout << "Q = " << Q << endl;
cout << "q2 = " << q2 << endl;
cout << "beta = " << beta << endl;
cout << "beta_q = " << beta_q << endl;
cout << "Exponents: ";
int_vec_print(cout, exponents, m);
cout << endl;
}
if (f_vv) {
cout << "twisted_tensor_product_codes: M:" << endl;
print_integer_matrix_width(cout, M, m, n, n, 2);
F->latex_matrix(cout, f_elements_exponential, symbol_for_print, M, m, n);
}
#if 0
for (j = 0; j < n; j++) {
PG_element_normalize(F, M + j, n, m);
}
cout << "column normalized M:" << endl;
print_integer_matrix_width(cout, M, m, n, n, 2);
#endif
C = NEW_int(m * m);
C_inv = NEW_int(m * m);
H_subfield = NEW_int(m * n);
create_matrix_H_subfield(F, f,
H_subfield, C, C_inv, M, m, n, beta, beta_q,
f_elements_exponential, symbol_for_print, symbol_for_print_subfield,
f_construction_A, f_hyperoval, f_construction_B,
verbose_level - 2);
if (f_v) {
cout << "twisted_tensor_product_codes: after create_matrix_H_subfield" << endl;
cout << "H_subfield:" << endl;
print_integer_matrix_width(cout, H_subfield, m, n, n, 2);
f->latex_matrix(cout, f_elements_exponential, symbol_for_print_subfield, H_subfield, m, n);
}
FREE_int(M);
FREE_int(C);
FREE_int(C_inv);
}
INT grassmann_embedded::rank_INT(INT *subspace_basis, INT verbose_level)
// subspace_basis is k x big_n
{
INT f_v = (verbose_level >= 1);
INT rk, i, j, a;
if (f_v) {
cout << "grassmann_embedded::rank_INT" << endl;
print_integer_matrix_width(cout, subspace_basis, k, big_n, big_n, F->log10_of_q);
}
for (i = 0; i < k; i++) {
for (j = 0; j < n; j++) {
a = subspace_basis[i * big_n + base_cols[j]];
tmp_M1[i * n + j] = a;
}
}
if (f_v) {
cout << "grassmann_embedded::rank_INT tmp_M1:" << endl;
print_integer_matrix_width(cout, tmp_M1, k, n, n, F->log10_of_q);
}
F->mult_matrix_matrix(tmp_M1, transform, tmp_M2, k, n, n);
if (f_v) {
cout << "grassmann_embedded::rank_INT tmp_M2:" << endl;
print_integer_matrix_width(cout, tmp_M2, k, n, n, F->log10_of_q);
}
for (i = 0; i < k; i++) {
F->mult_vector_from_the_left(tmp_M2 + i * n, M, Tmp2, n, big_n);
if (f_v) {
cout << "grassmann_embedded::rank_INT i=" << i << " Tmp2=" << endl;
print_integer_matrix_width(cout, Tmp2, 1, big_n, big_n, F->log10_of_q);
}
if (INT_vec_compare(subspace_basis + i * big_n, Tmp2, big_n)) {
cout << "grassmann_embedded::rank_INT fatal: the i-th vector is not in the space" << endl;
cout << "i=" << i << endl;
cout << "subspace:" << endl;
print_integer_matrix_width(cout, subspace_basis, k, big_n, big_n, F->log10_of_q);
cout << "space:" << endl;
print_integer_matrix_width(cout, M, n, big_n, big_n, F->log10_of_q);
cout << "Tmp1:" << endl;
INT_vec_print(cout, Tmp1, n);
cout << endl;
cout << "Tmp2:" << endl;
INT_vec_print(cout, Tmp2, big_n);
cout << endl;
exit(1);
}
for (j = 0; j < n; j++) {
G->M[i * n + j] = tmp_M2[i * n + j];
}
}
if (f_v) {
cout << "grassmann_embedded::rank_INT coefficient matrix:" << endl;
print_integer_matrix_width(cout, G->M, k, n, n, F->log10_of_q);
}
rk = G->rank_INT(verbose_level);
if (f_v) {
cout << "rk=" << rk << endl;
}
return rk;
}
void unusual_model::transform_matrix_unusual_to_usual(orthogonal *O,
INT *M4, INT *M5, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT f_vvv = (verbose_level >= 3);
INT *M4_tmp1, *M4_tmp2, /**M5,*/ *M5t, *M5_tmp1, *M5_tmp2;
INT i, j, a;
M4_tmp1 = NEW_INT(4 * 4);
M4_tmp2 = NEW_INT(4 * 4);
//M5 = NEW_INT(5 * 5);
M5t = NEW_INT(5 * 5);
M5_tmp1 = NEW_INT(5 * 5);
M5_tmp2 = NEW_INT(5 * 5);
if (f_v) {
cout << "unusual_model::transform_matrix_unusual_to_usual" << endl;
}
if (f_vv) {
cout << "transformation matrix in unusual model" << endl;
print_integer_matrix_width(cout, M4, 4, 4, 4, 3);
}
f.mult_matrix(hyperbolic_basis, M4, M4_tmp1, 4, 4, 4);
f.mult_matrix(M4_tmp1, hyperbolic_basis_inverse, M4_tmp2, 4, 4, 4);
if (f_vvv) {
cout << "transformation matrix in standard coordinates:" << endl;
print_integer_matrix_width(cout, M4_tmp2, 4, 4, 4, 3);
}
for (i = 0; i < 25; i++) {
M5[i] = 0;
}
for (i = 0; i < 4; i++) {
for (j = 0; j < 4; j++) {
a = M4_tmp2[i * 4 + j];
M5[(i + 1) * 5 + j + 1] = a;
}
}
M5[0 * 5 + 0] = 1;
if (f_vvv) {
cout << "embedded (M5):" << endl;
print_integer_matrix_width(cout, M5, 5, 5, 5, 3);
}
f.transpose_matrix(M5, M5t, 5, 5);
if (f_vvv) {
cout << "transposed (M5t):" << endl;
print_integer_matrix_width(cout, M5t, 5, 5, 5, 3);
cout << "Gram matrix:" << endl;
print_integer_matrix_width(cout, O->Gram_matrix, 5, 5, 5, 3);
}
f.mult_matrix(M5, O->Gram_matrix, M5_tmp1, 5, 5, 5);
f.mult_matrix(M5_tmp1, M5t, M5_tmp2, 5, 5, 5);
if (f_vvv) {
cout << "Gram matrix transformed:" << endl;
print_integer_matrix_width(cout, M5_tmp2, 5, 5, 5, 3);
}
for (i = 0; i < 25; i++) {
if (M5_tmp2[i] != O->Gram_matrix[i]) {
cout << "does not preserve the form" << endl;
exit(1);
}
}
#if 0
A->make_element(Elt, M5, verbose_level);
if (f_vv) {
A->print(cout, Elt);
}
#endif
FREE_INT(M4_tmp1);
FREE_INT(M4_tmp2);
//FREE_INT(M5);
FREE_INT(M5t);
FREE_INT(M5_tmp1);
FREE_INT(M5_tmp2);
}
void set_of_sets::intersection_matrix(
INT *&intersection_type, INT &highest_intersection_number,
INT *&intersection_matrix, INT &nb_big_sets,
INT verbose_level)
{
INT f_v = (verbose_level >= 1);
//INT f_vv = (verbose_level >= 2);
classify C;
INT f_second = FALSE;
INT f, l, a, i, ii, u, j;
if (f_v) {
cout << "set_of_sets::intersection_matrix" << endl;
}
C.init(Set_size, nb_sets, f_second, 0);
if (FALSE /*f_v*/) {
cout << "set_of_sets::intersection_matrix plane-intersection type: ";
C.print(FALSE /* f_backwards*/);
}
if (f_v) {
cout << "The intersection type is (";
C.print_naked(FALSE /* f_backwards*/);
cout << ")" << endl << endl;
}
f = C.type_first[C.nb_types - 1];
highest_intersection_number = C.data_sorted[f];
intersection_type = NEW_INT(highest_intersection_number + 1);
for (i = 0; i <= highest_intersection_number; i++) {
intersection_type[i] = 0;
}
for (i = 0; i < C.nb_types; i++) {
f = C.type_first[i];
l = C.type_len[i];
a = C.data_sorted[f];
intersection_type[a] = l;
}
f = C.type_first[C.nb_types - 1];
nb_big_sets = C.type_len[C.nb_types - 1];
INT *Incma, *Incma_t, *IIt, *ItI;
Incma = NEW_INT(underlying_set_size * nb_big_sets);
Incma_t = NEW_INT(nb_big_sets * underlying_set_size);
IIt = NEW_INT(underlying_set_size * underlying_set_size);
ItI = NEW_INT(nb_big_sets * nb_big_sets);
for (i = 0; i < underlying_set_size * nb_big_sets; i++) {
Incma[i] = 0;
}
for (i = 0; i < nb_big_sets; i++) {
ii = C.sorting_perm_inv[f + i];
for (j = 0; j < Set_size[ii]; j++) {
a = Sets[ii][j];
Incma[a * nb_big_sets + i] = 1;
}
}
if (FALSE /*f_vv*/) {
cout << "Incidence matrix:" << endl;
print_integer_matrix_width(cout, Incma, underlying_set_size, nb_big_sets, nb_big_sets, 1);
}
for (i = 0; i < underlying_set_size; i++) {
for (j = 0; j < underlying_set_size; j++) {
a = 0;
for (u = 0; u < nb_big_sets; u++) {
a += Incma[i * nb_big_sets + u] * Incma_t[u * underlying_set_size + j];
}
IIt[i * underlying_set_size + j] = a;
}
}
if (FALSE /*f_vv*/) {
cout << "I * I^\\top = " << endl;
print_integer_matrix_width(cout, IIt, underlying_set_size, underlying_set_size, underlying_set_size, 2);
}
for (i = 0; i < nb_big_sets; i++) {
for (j = 0; j < nb_big_sets; j++) {
a = 0;
for (u = 0; u < underlying_set_size; u++) {
a += Incma[u * nb_big_sets + i] * Incma[u * nb_big_sets + j];
}
ItI[i * nb_big_sets + j] = a;
}
}
if (FALSE /*f_v*/) {
cout << "I^\\top * I = " << endl;
print_integer_matrix_width(cout, ItI, nb_big_sets, nb_big_sets, nb_big_sets, 3);
}
intersection_matrix = NEW_INT(nb_big_sets * nb_big_sets);
for (i = 0; i < nb_big_sets; i++) {
for (j = 0; j < nb_big_sets; j++) {
intersection_matrix[i * nb_big_sets + j] = ItI[i * nb_big_sets + j];
}
}
FREE_INT(Incma);
FREE_INT(Incma_t);
FREE_INT(IIt);
FREE_INT(ItI);
if (f_v) {
cout << "set_of_sets::intersection_matrix done" << endl;
}
}
void unusual_model::setup2(INT q, const BYTE *poly_q, const BYTE *poly_Q, INT f_sum_of_squares, INT verbose_level)
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT f_vvv = (verbose_level >= 2);
INT Q, i, j, b, p, h;
if (f_v) {
cout << "unusual_model::setup q=" << q << " f_sum_of_squares=" << f_sum_of_squares << endl;
}
unusual_model::q = q;
Q = qq = q * q;
nb_terms = 0;
//const BYTE *override_poly_Q = NULL;
//const BYTE *override_poly_q = NULL;
is_prime_power(q, p, h);
#if 0
if (h > 1) {
override_poly_Q = override_polynomial_extension_field(q);
override_poly_q = override_polynomial_subfield(q);
F.init_override_polynomial(Q, override_poly_Q, verbose_level - 2);
cout << "field of order " << Q << " initialized" << endl;
f.init_override_polynomial(q, override_poly_q, verbose_level - 2);
}
else {
if (f_vv) {
cout << "initializing large field" << endl;
}
F.init(qq, verbose_level);
if (f_vv) {
cout << "initializing small field" << endl;
}
f.init(q, verbose_level);
if (f.e > 1) {
F.init(qq, 1);
f.init(q, 3);
cout << "need to choose the generator polynomial for the field" << endl;
F.compute_subfields(verbose_level);
exit(1);
}
}
#endif
if (f_vv) {
cout << "initializing large field" << endl;
}
F.init_override_polynomial(Q, poly_Q, verbose_level - 2);
if (f_vv) {
cout << "field of order " << Q << " initialized" << endl;
}
if (f_vv) {
cout << "initializing small field" << endl;
}
f.init_override_polynomial(q, poly_q, verbose_level - 2);
if (f_vv) {
cout << "field of order " << q << " initialized" << endl;
}
#if 0
if (q == 9) {
BYTE *override_poly_Q = "110"; // X^{4} + X^{3} + 2
BYTE *override_poly_q = "17"; // X^2 - X - 1 = X^2 +2X + 2 = 2 + 2*3 + 9 = 17
//finite_field::init_override_polynomial() GF(81) = GF(3^4), polynomial = X^{4} + X^{3} + 2 = 110
//subfields of F_{81}:
//subfield 3^2 : subgroup_index = 10
//0 : 0 : 1 : 1
//1 : 10 : 46 : X^{3} + 2X^{2} + 1
//2 : 20 : 47 : X^{3} + 2X^{2} + 2
F.init_override_polynomial(Q, override_poly_Q, verbose_level - 2);
cout << "field of order " << Q << " initialized" << endl;
f.init_override_polynomial(q, override_poly_q, verbose_level - 2);
}
else if (q == 25) {
BYTE *override_poly_Q = "767"; // X^{4} + X^{3} + 3X + 2
BYTE *override_poly_q = "47"; // X^2 - X - 3 = X^2 +4X + 2=25+20+2=47
//subfields of F_{625}:
//subfield 5^2 : subgroup_index = 26
//0 : 0 : 1 : 1
//1 : 26 : 110 : 4X^{2} + 2X
//2 : 52 : 113 : 4X^{2} + 2X + 3
F.init_override_polynomial(Q, override_poly_Q, verbose_level - 2);
cout << "field of order " << Q << " initialized" << endl;
f.init_override_polynomial(q, override_poly_q, verbose_level - 2);
}
else if (q == 27) {
BYTE *override_poly_Q = "974"; // X^{6} + X^{5} + 2
BYTE *override_poly_q = "34"; // X^3 - X + 1 = X^3 +2X + 1 = 27+6+1=34
//subfields of F_{729}:
//subfield 3^2 : subgroup_index = 91
//0 : 0 : 1 : 1
//1 : 91 : 599 : 2X^{5} + X^{4} + X^{3} + X + 2
//2 : 182 : 597 : 2X^{5} + X^{4} + X^{3} + X
//subfield 3^3 : subgroup_index = 28
//0 : 0 : 1 : 1
//1 : 28 : 158 : X^{4} + 2X^{3} + 2X^{2} + X + 2
//2 : 56 : 498 : 2X^{5} + X^{2} + X
//3 : 84 : 157 : X^{4} + 2X^{3} + 2X^{2} + X + 1
F.init_override_polynomial(Q, override_poly_Q, verbose_level - 2);
cout << "field of order " << Q << " initialized" << endl;
f.init_override_polynomial(q, override_poly_q, verbose_level - 2);
}
else if (q == 49) {
BYTE *override_poly_Q = "2754"; // X^{4} + X^{3} + X + 3
BYTE *override_poly_q = "94"; // X^2-X+3 = X^2+6X+3 = 49+6*7+3=94
//subfields of F_{2401}:
//subfield 7^2 : subgroup_index = 50
//0 : 0 : 1 : 1
//1 : 50 : 552 : X^{3} + 4X^{2} + X + 6
//2 : 100 : 549 : X^{3} + 4X^{2} + X + 3
F.init_override_polynomial(Q, override_poly_Q, verbose_level - 2);
cout << "field of order " << Q << " initialized" << endl;
f.init_override_polynomial(q, override_poly_q, verbose_level - 2);
}
else if (q == 81) {
BYTE *override_poly_Q = "6590"; // X^{8} + X^{3} + 2
BYTE *override_poly_q = "89"; // X^4-X-1=X^4+2X+2=81+2*3+2=89
//subfields of F_{6561}:
//subfield 3^4 : subgroup_index = 82
//0 : 0 : 1 : 1
//1 : 82 : 5413 : 2X^{7} + X^{6} + X^{5} + 2X^{3} + X^{2} + X + 1
//2 : 164 : 1027 : X^{6} + X^{5} + 2X^{3} + 1
//3 : 246 : 3976 : X^{7} + 2X^{6} + X^{5} + X^{4} + 2X + 1
//4 : 328 : 5414 : 2X^{7} + X^{6} + X^{5} + 2X^{3} + X^{2} + X + 2
F.init_override_polynomial(Q, override_poly_Q, verbose_level - 2);
cout << "field of order " << Q << " initialized" << endl;
f.init_override_polynomial(q, override_poly_q, verbose_level - 2);
}
else if (q == 121) {
BYTE *override_poly_Q = "15985"; // X^{4} + X^{3} + X + 2
BYTE *override_poly_q = "200"; // X^2-4X+2=X^2+7X+2=11^2+7*11+2=200
//subfields of F_{14641}:
//subfield 11^2 : subgroup_index = 122
//0 : 0 : 1 : 1
//1 : 122 : 4352 : 3X^{3} + 2X^{2} + 10X + 7
//2 : 244 : 2380 : X^{3} + 8X^{2} + 7X + 4
F.init_override_polynomial(Q, override_poly_Q, verbose_level - 2);
cout << "field of order " << Q << " initialized" << endl;
f.init_override_polynomial(q, override_poly_q, verbose_level - 2);
}
else {
}
#endif
alpha = f.p;
if (f_vv) {
cout << "primitive element alpha=" << alpha << endl;
}
if (f_vv) {
cout << "unusual_model::setup calling subfield_embedding_2dimensional" << endl;
}
F.subfield_embedding_2dimensional(f,
components, embedding, pair_embedding, verbose_level - 4);
if (f_vvv) {
cout << "unusual_model::setup subfield_embedding_2dimensional finished" << endl;
F.print_embedding(f, components, embedding, pair_embedding);
}
T_alpha = F.retract(f, 2, F.T2(alpha), verbose_level - 2);
N_alpha = F.retract(f, 2, F.N2(alpha), verbose_level - 2);
if (f_vv) {
cout << "T_alpha = " << T_alpha << endl;
cout << "N_alpha = " << N_alpha << endl;
}
form_i = NEW_INT(4 * 4);
form_j = NEW_INT(4 * 4);
form_coeff = NEW_INT(4 * 4);
Gram = NEW_INT(4 * 4);
for (i = 0; i < 4 * 4; i++)
Gram[i] = 0;
if (f_sum_of_squares) {
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 0, 0, 1);
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 1, 1, 1);
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 2, 2, 1);
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 3, 3, 1);
}
else {
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 0, 0, 1);
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 0, 1, T_alpha);
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 1, 1, N_alpha);
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 2, 2, 1);
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 2, 3, T_alpha);
add_term(4, F, nb_terms, form_i, form_j, form_coeff, Gram, 3, 3, N_alpha);
}
if (f_vv) {
cout << "Gram matrix:" << endl;
print_integer_matrix_width(cout, Gram, 4, 4, 4, 2);
cout << "quadratic form:" << endl;
print_quadratic_form_list_coded(nb_terms, form_i, form_j, form_coeff);
}
if (f_vv) {
cout << "finding hyperbolic pair" << endl;
}
f.find_hyperbolic_pair(4, nb_terms,
form_i, form_j, form_coeff, Gram,
basis, basis + 4, 0 /*verbose_level - 3*/);
f.perp(4, 2, basis, Gram);
if (f_vv) {
cout << "basis:" << endl;
print_integer_matrix_width(cout, basis, 4, 4, 4, 2);
}
for (i = 0; i < 2 * 4; i++) {
hyperbolic_basis[i] = basis[i];
}
if (f_vvv) {
for (i = 0; i < 4; i++) {
b = f.evaluate_quadratic_form(4, nb_terms, form_i, form_j, form_coeff,
basis + i * 4);
cout << "i=" << i << " form value " << b << endl;
}
}
f.restrict_quadratic_form_list_coding(4 - 2, 4, basis + 2 * 4,
nb_terms, form_i, form_j, form_coeff,
r_nb_terms, r_form_i, r_form_j, r_form_coeff,
verbose_level - 2);
if (f_vv) {
cout << "restricted quadratic form:" << endl;
print_quadratic_form_list_coded(r_nb_terms, r_form_i, r_form_j, r_form_coeff);
}
r_Gram = NEW_INT(2 * 2);
make_Gram_matrix_from_list_coded_quadratic_form(2, f,
r_nb_terms, r_form_i, r_form_j, r_form_coeff, r_Gram);
if (f_vv) {
cout << "restricted Gram matrix:" << endl;
print_integer_matrix_width(cout, r_Gram, 2, 2, 2, 2);
}
f.find_hyperbolic_pair(2, r_nb_terms,
r_form_i, r_form_j, r_form_coeff, r_Gram,
basis_subspace, basis_subspace + 2, verbose_level - 2);
if (f_vv) {
cout << "basis_subspace:" << endl;
print_integer_matrix_width(cout, basis_subspace, 2, 2, 2, 2);
}
f.mult_matrix(basis_subspace, basis + 8, hyperbolic_basis + 8, 2, 2, 4);
if (f_vv) {
cout << "hyperbolic basis:" << endl;
print_integer_matrix_width(cout, hyperbolic_basis, 4, 4, 4, 2);
for (i = 0; i < 4; i++) {
b = f.evaluate_quadratic_form(4, nb_terms, form_i, form_j, form_coeff, hyperbolic_basis + i * 4);
cout << "i=" << i << " quadratic form value " << b << endl;
}
}
M = NEW_INT(4 * 4);
for (i = 0; i < 4; i++) {
for (j = 0; j < 4; j++) {
M[i * 4 + j] = f.evaluate_bilinear_form(4, hyperbolic_basis + i * 4, hyperbolic_basis + j * 4, Gram);
}
}
if (f_vvv) {
cout << "bilinear form on the hyperbolic basis:" << endl;
print_integer_matrix_width(cout, M, 4, 4, 4, 2);
}
f.restrict_quadratic_form_list_coding(4, 4, hyperbolic_basis,
nb_terms, form_i, form_j, form_coeff,
rr_nb_terms, rr_form_i, rr_form_j, rr_form_coeff,
verbose_level - 2);
if (f_vv) {
cout << "restricted quadratic form:" << endl;
print_quadratic_form_list_coded(rr_nb_terms, rr_form_i, rr_form_j, rr_form_coeff);
}
f.matrix_inverse(hyperbolic_basis, hyperbolic_basis_inverse, 4, verbose_level - 2);
if (f_vv) {
cout << "inverse hyperbolic basis:" << endl;
print_integer_matrix_width(cout, hyperbolic_basis_inverse, 4, 4, 4, 2);
}
}
void coding_theory_domain::tt_field_reduction(
finite_field &F, finite_field &f,
int m, int n, int *M, int *MM, int verbose_level)
{
int f_v = (verbose_level >= 1);
int f_vv = (verbose_level >= 2);
int i, j, a, z, b, c, Q, q;
int index;
Q = F.q;
q = f.q;
index = (Q - 1) / (q - 1);
if (f_v) {
cout << "field reduction, Q=" << Q
<< " q=" << q << " index=" << index << endl;
}
if (f_vv) {
cout << "before:" << endl;
print_integer_matrix_width(cout, M, m, n, n, 2);
//print_integer_matrix(cout, M, m, n);
cout << endl;
F.print_integer_matrix_zech(cout, M, m, n);
cout << endl;
}
for (i = 0; i < m; i++) {
for (j = 0; j < n; j++) {
a = M[i * n + j];
if (a == 0) {
c = 0;
}
else {
if (f.e == 1) {
if (a >= q) {
cout << "field reduction: element does not "
"lie in the subfield: " << a << endl;
exit(1);
}
c = a;
}
else {
z = F.log_alpha(a);
b = z / index;
if (b * index != z) {
cout << "b * index != z" << endl;
exit(1);
}
c = f.alpha_power(b);
}
}
MM[i * n + j] = c;
}
}
if (f_vv) {
cout << "after:" << endl;
print_integer_matrix_width(cout, MM, m, n, n, 2);
//print_integer_matrix(cout, MM, m, n);
cout << endl;
f.print_integer_matrix_zech(cout, MM, m, n);
cout << endl;
}
}
void grassmann_embedded::init(INT big_n, INT n, grassmann *G, INT *M, INT verbose_level)
// M is n x big_n
{
INT f_v = (verbose_level >= 1);
INT f_vv = (verbose_level >= 2);
INT i, j, rk, idx;
longinteger_object deg;
longinteger_domain D;
grassmann_embedded::big_n = big_n;
grassmann_embedded::G = G;
grassmann_embedded::n = n;
if (G->n != n) {
cout << "grassmann_embedded::init n != G->n" << endl;
exit(1);
}
grassmann_embedded::k = G->k;
grassmann_embedded::F = G->F;
grassmann_embedded::q = G->F->q;
if (f_v) {
cout << "grassmann_embedded::init big_n = " << big_n << " n=" << n << " k=" << k << " q=" << q << endl;
}
base_cols = NEW_INT(big_n);
embedding = NEW_INT(big_n);
grassmann_embedded::M = NEW_INT(n * big_n);
M_Gauss = NEW_INT(n * big_n);
transform = NEW_INT(n * n);
tmp_M1 = NEW_INT(n * n);
tmp_M2 = NEW_INT(n * n);
Tmp1 = NEW_INT(big_n);
Tmp2 = NEW_INT(big_n);
Tmp3 = NEW_INT(big_n);
for (i = 0; i < n * big_n; i++) {
grassmann_embedded::M[i] = M[i];
M_Gauss[i] = M[i];
}
// we initialize transform as the identity matrix:
for (i = 0; i < n * n; i++) {
transform[i] = 0;
}
for (i = 0; i < n; i++) {
transform[i * n + i] = 1;
}
if (f_vv) {
cout << "grassmann_embedded::init subspace basis before Gauss reduction:" << endl;
print_integer_matrix_width(cout, grassmann_embedded::M, n, big_n, big_n, F->log10_of_q);
}
//rk = F->Gauss_simple(M_Gauss, n, big_n, base_cols, verbose_level - 1);
rk = F->Gauss_INT(M_Gauss, FALSE /*f_special*/, TRUE/* f_complete*/, base_cols,
TRUE /* f_P */, transform, n, big_n, n, 0/*INT verbose_level*/);
if (f_vv) {
cout << "grassmann_embedded::init subspace basis after reduction:" << endl;
print_integer_matrix_width(cout, M_Gauss, n, big_n, big_n, F->log10_of_q);
cout << "grassmann_embedded::init transform:" << endl;
print_integer_matrix_width(cout, transform, n, n, n, F->log10_of_q);
}
if (f_v) {
cout << "base_cols:" << endl;
INT_vec_print(cout, base_cols, rk);
cout << endl;
}
if (rk != n) {
cout << "grassmann_embedded::init rk != n" << endl;
cout << "rk=" << rk << endl;
cout << "n=" << n << endl;
exit(1);
}
j = 0;
for (i = 0; i < big_n; i++) {
if (!INT_vec_search(base_cols, n, i, idx)) {
embedding[j++] = i;
}
}
if (j != big_n - n) {
cout << "j != big_n - n" << endl;
cout << "j=" << j << endl;
cout << "big_n - n=" << big_n - n << endl;
exit(1);
}
if (f_v) {
cout << "embedding: ";
INT_vec_print(cout, embedding, big_n - n);
cout << endl;
}
D.q_binomial(deg, n, k, q, 0);
degree = deg.as_INT();
}
void coding_theory_domain::create_matrix_M(
int *&M,
finite_field *F, finite_field *f,
int &m, int &n, int &beta, int &r, int *exponents,
int f_construction_A, int f_hyperoval, int f_construction_B,
int f_elements_exponential, const char *symbol_for_print,
int verbose_level)
// int exponents[9]
{
int f_v = (verbose_level >= 1);
//int f_vv = (verbose_level >= 2);
int i, j, t, q, Q, q2;
q = f->q;
q2 = q * q;
if (f_construction_A) {
#if 0
//q = 2; override_poly_Q = ""; override_poly = "";
//q = 3; override_poly_Q = ""; override_poly = "";
q = 4; override_poly_Q = "19"; override_poly = "7";
// F_256 generated by X^8 + X^4 + X^3 + X^2 + 1
// F_16 generated by X^4+X+1 = 19
// F_4 generated by X^2+X+1 = 7
//q = 5; override_poly_Q = "47"; override_poly = "";
// F_625 generated by X^4 + X^3 + 3X + 2
// F_25 generated by X^2 + 4X + 2 = 47
//q = 7; override_poly_Q = ""; override_poly = "";
//q = 8; override_poly_Q = "97"; override_poly = "11";
// F_4096 generated by x^12+x^6+x^4+x+1
// F_64 generated by X^6+X^5+1 = 97
// F_8 generated by X^3+X+1 = 11
//q = 9; override_poly_Q = ""; override_poly = "17";
#endif
Q = q2;
beta = q;
m = 9;
if (f_hyperoval) {
n = Q + 2;
}
else {
n = Q + 1;
}
r = 5;
if (q == 4)
r = 7;
if (q == 3)
r = 9;
// 3 orbits of length 1: 0, q+1, 2q+2
exponents[0] = 0;
exponents[1] = q + 1;
exponents[2] = 2 * q + 2;
//orbit (q,1)
exponents[3] = q;
exponents[4] = 1;
// orbit (2q, 2)
exponents[5] = 2 * q;
exponents[6] = 2;
// orbit (2q+1, q+2)
exponents[7] = 2 * q + 1;
exponents[8] = q + 2;
//exponents[0] = 0;
//exponents[1] = q;
//exponents[2] = 2 * q;
//exponents[3] = 1;
//exponents[4] = q + 1;
//exponents[5] = 2 * q + 1;
//exponents[6] = 2;
//exponents[7] = q + 2;
//exponents[8] = 2 * q + 2;
}
else if (f_construction_B) {
#if 0
//q = 2; override_poly_Q = ""; override_poly = ""; r = 9;
//q = 3; override_poly_Q = ""; override_poly = ""; r = 5;
q = 4; override_poly_Q = ""; override_poly = "7"; r = 4;
// F_4096 generated by X^8 + X^4 + X^3 + X^2 + 1 = 4096
//q = 5; override_poly_Q = ""; override_poly = ""; r = 4;
//q = 7; override_poly_Q = ""; override_poly = ""; r = 4;
#endif
beta = q;
Q = q2 * q;
m = 8;
n = Q + 1;
exponents[0] = 0;
exponents[1] = q2 + q + 1;
exponents[2] = 1;
exponents[3] = q;
exponents[4] = q2;
exponents[5] = q + 1;
exponents[6] = q2 + q;
exponents[7] = q2 + 1;
//exponents[0] = 0;
//exponents[1] = q;
//exponents[2] = 1;
//exponents[3] = q + 1;
//exponents[4] = q2;
//exponents[5] = q2 + q;
//exponents[6] = q2 + 1;
//exponents[7] = q2 + q + 1;
}
else {
cout << "create_matrix_M(): please specify the construction using option -A or -B" << endl;
exit(1);
}
// create matrix M:
M = NEW_int(m * n);
for (i = 0; i < m; i++) {
for (j = 0; j < n; j++) {
M[i * n + j] = 0;
}
}
for (t = 0; t < Q; t++) {
for (i = 0; i < m; i++) {
M[i * n + t] = F->power(t, exponents[i]);
}
}
if (f_construction_A) {
M[2 * n + Q] = 1;
if (f_hyperoval)
M[1 * n + Q + 1] = 1;
}
else if (f_construction_B) {
M[1 * n + Q] = 1;
}
if (f_v) {
cout << "M:" << endl;
print_integer_matrix_width(cout, M, m, n, n, 2);
F->latex_matrix(cout, f_elements_exponential,
symbol_for_print, M, m, n);
}
if (f_v) {
int *all_one, *col_sum;
all_one = NEW_int(n);
col_sum = NEW_int(m);
for (i = 0; i < n; i++)
all_one[i] = 1;
F->mult_matrix_matrix(M, all_one, col_sum, m, n, 1,
0 /* verbose_level */);
cout << "overall col_sum:" << endl;
print_integer_matrix_width(cout, col_sum, m, 1, 1, 2);
FREE_int(all_one);
FREE_int(col_sum);
}
}