Construction of a Plane#

All functions which create a plane return three values:

(i): the plane itself;
(ii): the point–set of the plane;
(iii): the line–set of the plane.

These “sets” ((ii) and (iii)) are used as the parent structures for points and lines respectively, and are explained more fully in the next section.

FiniteProjectivePlane< v | X : parameters >: RngIntElt, List -> PlaneProj#

FiniteProjectivePlane< V | X : parameters >: SetIndx, List -> PlaneProj#

Check: BoolElt                    Default: true

Construct the projective plane \(P\) having as point set the indexed set \(V\) (or \(\lbrace @ 1, 2, \ldots, v @ \rbrace\) if an integer \(v\) is given), and as line set \(L = \lbrace L_1, L_2, \ldots, L_b \rbrace\) given by the list \(X\). The value of \(X\) must be either:

(a): A list of subsets of the set \(V\).
(b): A sequence, set or indexed set of subsets of \(V\).
(c): A list of lines of an existing plane.
(d): A sequence, set or indexed set of lines of an existing plane.
(e): A combination of the above.
(f): A \(v \times b\) \((0,1)\)-matrix \(A\), where \(A\) may be defined over any coefficient ring. The matrix \(A\) will be interpreted as the incidence matrix for the plane \(P\).
(g): A set of codewords of a linear code with length \(v\). The line set of \(P\) is taken to be the set of supports of the codewords.

The optional boolean argument Check indicates whether or not to check that the given data satisfies the projective plane axioms.

FiniteProjectivePlane(W): ModTupFld -> PlaneProj#
FiniteProjectivePlane(F): FldFin -> PlaneProj#
FiniteProjectivePlane(q): RngIntElt -> PlaneProj#: Given a \(3\)–dimensional vector space \(W\) defined over the field \(F = {\bf F}_{q}\), construct the classical projective plane defined by the one–dimensional and two–dimensional subspaces of \(W\).

FiniteAffinePlane< v | X : parameters >: RngIntElt, List -> PlaneAff#

FiniteAffinePlane< V | X : parameters >: SetIndx, List -> PlaneAff#

Check: BoolElt                    Default: true

Construct the affine plane \(P\) having as point set the indexed set \(V\) (or \(\lbrace @ 1, 2, \ldots, v @ \rbrace\) if an integer \(v\) is given), and as line set \(L = \lbrace L_1, L_2, \ldots, L_b \rbrace\) given by the list \(X\). The value of \(X\) must be either:

(a): A list of subsets of the set \(V\).
(b): A sequence, set or indexed set of subsets of \(V\).
(c): A list of lines of an existing plane.
(d): A sequence, set or indexed set of lines of an existing plane.
(e): A combination of the above.
(f): A \(v \times b\) \((0,1)\)-matrix \(A\), where \(A\) may be defined over any coefficient ring. The matrix \(A\) will be interpreted as the incidence matrix for the plane \(P\).
(g): A set of codewords of a linear code with length \(v\). The line set of \(P\) is taken to be the set of supports of the codewords.

The optional boolean argument Check indicates whether or not to check that the given data satisfies the affine plane axioms.

FiniteAffinePlane(W): ModFld -> PlaneAff#
FiniteAffinePlane(F): FldFin -> PlaneProj#
FiniteAffinePlane(q): RngIntElt -> PlaneProj#: Given a \(2\)–dimensional vector space \(W\) defined over the field \(F = {\bf F}_{q}\), construct the classical affine plane defined by the cosets of the subspaces of \(W\).

Example: Constructors (ex-9ba67e)#

The classical projective plane of order 3 can be constructed by the following statement:

> P, V, L := FiniteProjectivePlane(3);
> P;
Projective Plane PG(2, 3)
> V;
Point-set of Projective Plane PG(2, 3)
> L;
Line-set of Projective Plane PG(2, 3)

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A non-classical affine plane of order 2 can be constructed in the following way:

> A := FiniteAffinePlane< 4 | Setseq(Subsets({1, 2, 3, 4}, 2)) >;
> A: Maximal;
Affine Plane of order 2
Points: {@ 1, 2, 3, 4 @}
Lines:
    {1, 3},
    {1, 4},
    {2, 4},
    {2, 3},
    {1, 2},
    {3, 4}

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To demonstrate the use of the Check argument, we recreate the classical projective plane of order 16 with Check := true (the default) and Check := false.

> P, V, L := FiniteProjectivePlane(16);
> time P2 := FiniteProjectivePlane<
>     Points(P) | {Set(l): l in L} : Check := true >;
Time: 10.769
> time P2 := FiniteProjectivePlane<
>     Points(P) | {Set(l): l in L} : Check := false >;
Time: 0.030

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