General Design Constructions

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 \(D\).

The Construction of Related Structures

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 \(D\).

Dual(D): Inc -> Inc

The dual of the incidence structure \(D\).

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

Given an incidence structure \(D = (P, B)\), and a point \(p \in P\), form the incidence structure

\[E = ( P - \lbrace p \rbrace, \lbrace b - \lbrace p \rbrace : b \in B | p \in b \rbrace ).\]

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

\[E = ( b, \lbrace b \cap c : c \in B | c \ne b \rbrace ).\]

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.

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

Given an incidence structure \(D = (P, B)\), and a block \(b \in B\), form the incidence structure \(E = ( P - b, B - \lbrace b \rbrace ).\) 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

\[E = ( P - \lbrace p \rbrace, \lbrace x : x \in B | p \notin x \rbrace ).\]

Thus, \(E\) has point set \(P - \lbrace p \rbrace\) and its blocks are the blocks of \(D\) which do not contain \(p\).

Simplify(D): Inc -> Inc

Simplify the incidence structure \(D\); i.e., remove repeated blocks from \(D\).

Sum(Q): [ Inc ] -> Inc

Given a sequence \(Q = [ D_1, \ldots, D_l ]\) 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 \(D_1, \ldots, D_l\). Thus, if \(D_i = (P, B_i)\) then \(D = (P, B_1 \cup \ldots \cup B_l)\).

Union(D, E): Inc, Inc -> Inc

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

Restriction(D, S): IncNsp, { Incpt } -> IncNsp

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

Example: Related Structures of a Design (ex-2db350)

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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}

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The Witt Designs

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 \(24\).

Example: Contraction of a Witt Design (ex-9c4643)

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

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Difference Sets and their Development

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

\[\lbrace gh^{-1} : g, h \in D | g \neq h \rbrace\]

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 [Hal86], pp. 141–142.

SingerDifferenceSet(n, q): RngIntElt, RngIntElt -> { RngIntResElt }

The Singer difference set corresponding to a hyperplane of \(\operatorname{PG}(n, q)\).

IsDifferenceSet(B): SetEnum -> BoolElt, RngIntElt

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 \(\lambda\) (i.e., the number of times each non–identity group/ring element appears as a “difference” of elements of \(B\)) is also returned.

Development(B): {  RngElt } -> Inc

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 \(T = \lbrace B_1, \ldots, B_l \rbrace\) be a difference family consisting of subsets of a magma \(A\) which is either the ring \(\Z/m\Z\), 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 \(\lbrace B_1 \cup \ldots \cup B_l \rbrace\) translated by the \(i\)-th element of \(A\).

Example: Developing a Difference Set (ex-b4f959)

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The set \(\lbrace 1, 3, 4, 5, 9 \rbrace\), 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 (\(= 17 \times 19\)), and its development.

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

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