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Each of these functions returns three values:
- (i)
- The incidence structure D;
- (ii)
- The point-set P of D;
- (iii)
- The block-set B of D.
All operations defined for incidence structures apply also
to near--linear spaces, linear spaces and designs.
The complement of the incidence structure D.
The dual of the incidence structure D.
Given an incidence structure D = (P, B), and a point p ∈P,
form the incidence structure
E = ( P - { p }, { b - { p } : b ∈B | p ∈b } ).
Thus, E is constructed from D by deleting p and retaining only
those blocks incident with it.
Given an incidence structure D = (P, B), and a block b ∈B,
form the incidence structure
E = ( b, { b ∩c : c ∈B | c ≠b } ).
Thus, E has point set b and its blocks are the non--empty
intersections of b with the blocks of D other than b itself.
Given an incidence structure D = (P, B), and a block b ∈B,
form the incidence structure E = ( P - b, B - { b } ).
Thus, E has point set P - b and its blocks are the non--empty
intersections of P - b with the blocks of D.
Given an incidence structure D = (P, B), and a point p ∈P,
form the incidence structure
E = ( P - { p }, { x : x ∈B | p ∉x } ).
Thus, E has point set P - { p } and its blocks are
the blocks of D which do not contain p.
Simplify the incidence structure D; i.e., remove repeated blocks
from D.
Given a sequence Q = [ D1, ..., Dl ] of incidence structures,
each of which is defined over the same set P of points,
form the incidence structure obtained by taking the union of the
block sets of D1, ..., Dl. Thus, if Di = (P, Bi)
then D = (P, B1 ∪ ... ∪Bl).
The union of incidence structures D and E.
That is, if D = (P, B) and E = (Q, C), then return
U = (P ∪Q, B ∪C).
The point sets P and Q must be disjoint.
The restriction of the (near--)linear space D to the set of points S.
We illustrate some of the above functions with an example.
> K := Design< 3, 8 | {1,3,7,8}, {1,2,4,8}, {2,3,5,8}, {3,4,6,8}, {4,5,7,8},
> {1,5,6,8}, {2,6,7,8}, {1,2,3,6}, {1,2,5,7}, {1,3,4,5}, {1,4,6,7}, {2,3,4,7},
> {2,4,5,6}, {3,5,6,7} >;
> CK := Contraction(K, Point(K, 8));
> RK := Residual(K, Block(K, 1));
> K: Maximal;
3-(8, 4, 1) Design with 14 blocks
Points: {@ 1, 2, 3, 4, 5, 6, 7, 8 @}
Blocks:
{1, 3, 7, 8},
{1, 2, 4, 8},
{2, 3, 5, 8},
{3, 4, 6, 8},
{4, 5, 7, 8},
{1, 5, 6, 8},
{2, 6, 7, 8},
{1, 2, 3, 6},
{1, 2, 5, 7},
{1, 3, 4, 5},
{1, 4, 6, 7},
{2, 3, 4, 7},
{2, 4, 5, 6},
{3, 5, 6, 7}
> CK: Maximal;
2-(7, 3, 1) Design with 7 blocks
Points: {@ 1, 2, 3, 4, 5, 6, 7 @}
Blocks:
{1, 3, 7},
{1, 2, 4},
{2, 3, 5},
{3, 4, 6},
{4, 5, 7},
{1, 5, 6},
{2, 6, 7}
> RK: Maximal;
Incidence Structure on 4 points with 13 blocks
Points: {@ 2, 4, 5, 6 @}
Blocks:
{2, 4},
{2, 5},
{4, 6},
{4, 5},
{5, 6},
{2, 6},
{2, 6},
{2, 5},
{4, 5},
{4, 6},
{2, 4},
{2, 4, 5, 6},
{5, 6}
> RKS := Simplify(RK);
> RKS: Maximal;
Incidence Structure on 4 points with 7 blocks
Points: {@ 2, 4, 5, 6 @}
Blocks:
{2, 4},
{2, 5},
{4, 6},
{4, 5},
{5, 6},
{2, 6},
{2, 4, 5, 6}
The 5--(12, 6, 1) and 5--(24, 8, 1) designs constructed
by Witt, also
known as the small and large Mathieu designs, respectively, can
be constructed in Magma with the following function.
The Witt 5--design on n points, where n = 12 or 24.
We construct the Witt 5--(24, 8, 1) design and take its contraction at
a point. This contraction is in fact isomorphic to the
design constructed above from the unextended binary Golay code.
> D, P, B := WittDesign(24);
> D;
5-(24, 8, 1) Design with 759 blocks
> p := P.1;
> Cp := Contraction(D, p);
> Cp;
4-(23, 7, 1) Design with 253 blocks
Let G be a group of order v and let k and λ be positive
integers such that 1 < k < v. A (v, k, λ) difference set
for G is a set D of k group elements such that the set
{ gh - 1 : g, h ∈D | g != h }
contains every non--identity element of G exactly λ times.
The difference set of type given by t (which must be one of
"Q", "H6", "T", "B",
"B0", "O", "O0", or "W4")
corresponding to the prime p.
The types have the same interpretation as given by Marshall Hall
in [Hal86], pp. 141--142.
The Singer difference set corresponding to a hyperplane of PG(n, q).
Returns true iff B is a difference set over an integer residue class ring
or a finite group (with an iterator). If true, the value of the parameter λ
(i.e., the number of times each non--identity group/ring element appears
as a "difference" of elements of B) is also returned.
Let B be a subset of a magma A which is a difference set relative
to A, where A is either the ring Z/mZ, a finite abelian group or
an arbitrary finite group (with an iterator).
This function constructs the symmetric design
having point set A and whose blocks consist of the sets obtained by
translating B by each element of A in turn.
Let T = { B1, ..., Bl } be a difference family consisting
of subsets of a magma A which is either the ring Z/mZ, a finite
abelian group or an arbitrary finite group (with an iterator).
This function constructs the
incidence structure with point set A and whose i-th block is the set
{ B1 ∪ ... ∪Bl } translated by the i-th element
of A.
The set { 1, 3, 4, 5, 9 }, where the elements are
residues modulo 11, forms an (11, 5, 2) difference set. We develop
this set and construct a 2-(11, 5, 2) design.
> Z11 := IntegerRing(11);
> B := { Z11 | 1, 3, 4, 5, 9};
> IsDifferenceSet(B);
true 2
> D := Development(B);
> D: Maximal;
2-(11, 5, 2) Design with 11 blocks
Points: {@ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 @}
Blocks:
{1, 3, 4, 5, 9},
{2, 4, 5, 6, 10},
{0, 3, 5, 6, 7},
{1, 4, 6, 7, 8},
{2, 5, 7, 8, 9},
{3, 6, 8, 9, 10},
{0, 4, 7, 9, 10},
{0, 1, 5, 8, 10},
{0, 1, 2, 6, 9},
{1, 2, 3, 7, 10},
{0, 2, 3, 4, 8}
We now construct the twin primes (type "T") difference set modulo 323 (= 17 x 19), and its development.
> B := DifferenceSet(17, "T");
> D := Development(B);
> D;
2-(323, 161, 80) Design with 323 blocks
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