Construction of BIBD’s Using Quadratic Residues

Definition: Let v, k, and λ be positive integers such that v > k ≥ 2. A (v, k, λ)-balanced incomplete block design (which we abbreviate to (v, k, λ)-BIBD) is a design (X,A) such that the following properties are satisfied: 1. |X| = v, 2. Each block contains exactly k points, and 3. Every pair of distinct points is contained in exactly λ blocks. Property 3 in the definition above is the “balance” property. A BIBD iscalled an incomplete block design because k < v, and hence all its blocks are incomplete blocks. A BIBD may possibly contain repeated blocks if λ > 1 Example: 1. A (7, 3, 1)-BIBD. X = {1, 2, 3, 4, 5, 6, 7}, and A = {123, 145, 167, 246, 257, 347, 356}. The blocks of the BIBD are the six lines and the circle in this diagram.