• A -Design is an incidence structure

    More specifically, a design is a set of points, with a collection of -subsets of points called blocks such that every -set of points lies in precisely blocks

  • The projective planes have the property that every two points lie in a unique block.

    They are therefore designs

  • Let be a design and a set of points where .

    We have

    So that is also an design.

  • A -design exists if is an integer for all .

  • denotes the total blocks of the design.

  • is the replication number which denotes the number of blocks containing each point.

  • (Godsil 5.10.1) In a -design with , we have .

  • A -design with is called symmetric.

  • (Godsil 5.10.2) The dual of a symmetric design is a symmetric design with the same parameters

  • (Godsil 5.10.3) A bipartite graph is the incidence graph of a symmetric -design if and only if it is distance regular with diameter

  • The incidence matrix of a design is the matrix where rows are indexed by points, columns by blocks.

    if the -th block in the -th block.

  • satisfies

    Where is the all one’s matrix.

Steiner system

  • A Steiner System is one where .

  • A Steiner Triple system is a -design with , and .

  • (Godsil 5.10.4) The block intersection graph of a Steiner triple system with is distance regular with diameter .

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