of size 9; a 2-(37,9,2)). There are five biplanes of order 9 (and 56 points, lines of size 11; a 2-(56,11,2)). Two biplanes are known of order 11 (and 79 points, lines of size 13; a 2-(79,13,2)). Biplanes of orders 5, 6, 8 and 10 do not exist, as shown by the Bruck-Ryser-Chowla theorem. Hadamard 2-designs An Hadamard matrix of size m is an m × m matrix H whose entries are ±1 such that HH⊤ = mIm, where H⊤ is the transpose of H and Im is the m × m identity matrix. An Hadamard matrix can be put into standardized form (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size m > 2 then m must be a multiple of 4. Given an Hadamard matrix of size 4a in standardized form, remove the first row and first column and convert every −1 to a 0. The resulting 0–1 matrix M is the incidence matrix of a symmetric 2-(4a − 1, 2a − 1, a − 1) design called an Hadamard 2-design. It contains blocks/points; each contains/is contained in points/blocks. Each pair of points is contained in exactly blocks. This construction is reversible, and the incidence matrix of a symmetric 2-design with these parameters can be used to form an Hadamard matrix of size 4a. Resolvable 2-designs A resolvable 2-design is a BIBD whose blocks can be partitioned into sets (called parallel classes), each of which forms a partition of the point set of the BIBD. The set of parallel classes is called a resolution of the design. If a 2-(v,k,λ) resolvable design has c parallel classes, then b ≥ v + c − 1. Consequently, a symmetric design can not have a non-trivial (more than one parallel class) resolution. Archetypical resolvable 2-designs are the finite affine planes. A solution of the famous 15 schoolgirl problem is a resolution of a 2-(15,3,1) design. General balanced designs (t-designs) Given any positive integer t, a t-design B is a class of k-element subsets of X, called blocks, such that every point x in X appears in exactly r blocks, and every t-element subset T appears in exactly λ blocks. The numbers v (the number of elements of X), b (the number of blocks), k, r, λ, and t are the parameters of the design. The design may be