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));
> 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.

WittDesign(n) : RngIntElt -> Dsgn

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

> 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

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];
> IsHadamard(H);
true
> DR := HadamardRowDesign(H, 3);
> 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);
> 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 types have the same interpretation as given by Marshall Hall in Combinatorial Theory (New York: Wiley, 2nd ed. 1986), pp. 141--142.

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 (with an iterator). 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 (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.

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 (with an iterator). 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};
> 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}

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