of (repeated subsets are allowed). Normally a block design is required to satisfy numerical regularity conditions. As an incidence structure, is the set of points and is the set of lines, usually called blocks in this context (repeated blocks must have distinct names, so is actually a set and not a multiset). If all the subsets in have the same size, the block design is called uniform. If each element of appears in the same number of subsets, the block design is said to be regular. The dual of a uniform design is a regular design and vice versa. Example: Fano plane Consider the block design/hypergraph given by: This incidence structure is called the Fano plane. As a block design it is both uniform and regular. In the labeling given, the lines are precisely the subsets of the points that consist of three points whose labels add up to zero using nim addition. Alternatively, each number, when written in binary, can be identified with a non-zero vector of length three over the binary field. Three vectors that generate a subspace form a line; in this case, that is equivalent to their vector sum being the zero vector. Representations Incidence structures may be represented in many ways. If the sets and are finite these representations can compactly encode all the relevant information concerning the structure. Incidence matrix The incidence matrix of a (finite) incidence structure is a (0,1) matrix that has its rows indexed by the points } and columns indexed by the lines } where the ij-th entry is a 1 if and 0 otherwise. An incidence matrix is not uniquely determined since it depends upon the arbitrary ordering of the points and the lines. The non-uniform incidence structure pictured above (example number 2) is given by: An incidence matrix for this structure is: which corresponds to the incidence table: If an incidence structure has an incidence matrix , then the dual structure has the transpose matrix T as its incidence matrix (and is defined by that matrix). An incidence structure is self-dual if there exists an ordering of the points and lines so that the incidence matrix constructed with that ordering is a symmetric matrix. With the labels as given in example number 1 above and with points ordered and lines ordered , the Fano plane has the incidence matrix: Since this is a symmetric matrix, the Fano plane