Each of these functions returns three values:

• The incidence structure D;
• The point-set P of D;
• The block-set B of D.

Subsections

• The Construction of Related Structures
• The Witt Designs
• Hadamard Matrices and their 3-Designs
• Difference Sets and their Development

All operations defined for incidence structures apply also to near-linear spaces, linear spaces and designs.

Complement(D) : Inc -> Inc

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 in P, form the incidence structure E = ( P - { p }, { b - { p } : b in B | p in b } ). Thus, E is constructed from D by deleting p and retaining only those blocks incident with it.

Contraction(D, b) : Inc, IncBlk -> Inc

Given an incidence structure D = (P, B), and a block b in B, form the incidence structure E = ( b, { b intersect c : c in 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 in 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.

Residual(D, p) : Inc, IncPt -> Inc

Given an incidence structure D = (P, B), and a point p in P, form the incidence structure E = ( P - { p }, { x : x in B | p notin 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 = [ D_1, ..., D_l ] 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 D_1, ..., D_l. Thus, if D_i = (P, B_i) then D = (P, B_1 union ... union B_l).

The union of incidence structures D and E. That is, if D = (P, B) and E = (Q, C), then return U = (P union Q, B union C). The point sets P and Q must be disjoint.

The restriction of the (near-)linear space D to the set of points S.

> 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));
> print 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}
> print 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}
> print 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);
> print 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.

WittDesign(n) : RngIntElt -> Dsgn

The Witt 5-design on n points, where n = 12 or 24.

> D, P, B := WittDesign(24); 
> print D;
5-(24, 8, 1) Design with 759 blocks
> p := P.1;
> Cp := Contraction(D, p);
> print Cp;
4-(23, 7, 1) Design with 253 blocks

Many designs can be constructed from Hadamard matrices. Magma provides functions to create such designs, as well as functions to test if a matrix is Hadamard or if two Hadamard matrices are Hadamard equivalent.

IsHadamard(H) : AlgMatElt -> BoolElt

True iff H is a Hadamard matrix. H may be defined over any ring.

True iff the Hadamard matrices H and J are Hadamard equivalent.

Given an n x n Hadamard matrix H (with n >= 4) and an integer i such that 1 <= i <= n, return the Hadamard 3-design corresponding to the row i of H.

Given an n x n Hadamard matrix H (with n >= 4) and an integer i such that 1 <= i <= n, return the Hadamard 3-design corresponding to the column i of H.

> R := MatrixRing(Integers(), 8);
> H := R![1, 1, 1, 1, 1, 1, 1, 1,
>       1, 1, 1, 1, -1, -1, -1, -1,
>       1, 1, -1, -1, 1, 1, -1, -1,
>       1, 1, -1, -1, -1, -1, 1, 1,
>       1, -1, 1, -1, 1, -1, -1, 1,
>       1, -1, 1, -1, -1, 1, 1, -1,
>       1, -1, -1, 1, -1, 1, -1, 1,
>       1, -1, -1, 1, 1, -1, 1, -1];
> print IsHadamard(H);
true
> DR := HadamardRowDesign(H, 3);
> print DR: Maximal;
3-(8, 4, 1) Design with 14 blocks
Points: {@ 1, 2, 3, 4, 5, 6, 7, 8 @}
Blocks:
    {1, 2, 5, 6},
    {1, 2, 7, 8},
    {1, 2, 3, 4},
    {1, 4, 5, 7},
    {1, 4, 6, 8},
    {1, 3, 6, 7},
    {1, 3, 5, 8},
    {3, 4, 7, 8},
    {3, 4, 5, 6},
    {5, 6, 7, 8},
    {2, 3, 6, 8},
    {2, 3, 5, 7},
    {2, 4, 5, 8},
    {2, 4, 6, 7}
> DC := HadamardColumnDesign(H, 8);
> print DC;
3-(8, 4, 1) Design with 14 blocks

Let G be a group of order v and let k and lambda be positive integers such that 1 < k < v. A (v, k, lambda) difference set for G is a set D of k group elements such that the set { gh^(-1) : g, h in D | g != h } contains every non-identity element of G exactly lambda times.

DifferenceSet(p, t) : RngIntElt, MonStgElt -> { RngIntResElt }

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 Singer difference set corresponding to a hyperplane of PG(n, q).

True iff B is a difference set over an integer residue class ring or a finite group. If true, the value of the parameter lambda (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. This function constructs the symmetric design D having point set A and whose blocks consist of the sets obtained by translating B by each element of A in turn.

Development(T) : { { Elt } } -> Inc

Let T = { B_1, ..., B_l } 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. This function constructs the incidence structure with point set A and whose i-th block is the set { B_1 union ... union B_l } translated by the i-th element of A.

> Z11 := IntegerRing(11);
> B := { Z11 | 1, 3, 4, 5, 9};
> print IsDifferenceSet(B);
true 2
> D := Development(B);
> print 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}

> B := DifferenceSet(17, "T");
> D := Development(B);
> print D;
2-(323, 161, 80) Design with 323 blocks