Mathematical Background

In order to be able to read this book a fairly thorough mathematical background is necessary. In different chapters many different areas of mathematics play a rôle. The most important one is certainly algebra but the reader must also know some facts from elementary number theory, probability theory and a number of concepts from combinatorial theory such as designs and geometries. In the following sections we shall give a brief survey of the prerequisite knowledge. Usually proofs will be omitted. For these we refer to standard textbooks. In some of the chapters we need a large number of facts concerning a not too well-known class of orthogonal polynomials, called Krawtchouk polynomials. These properties are treated in Section 1.2. The notations that we use are fairly standard. We mention a few that may not be generally known. If C is a finite set we denote the number of elements of C by ∣C∣. If the expression B is the definition of concept A then we write A := B. We use “iff” for “if and only if”. An identity matrix is denoted by I and the matrix with all entries equal to one is J. Similarly we abbreviate the vector with all coordinates 0 (resp. 1) by 0 (resp. 1). Instead of using [x] we write ⌊x⌋ := max{n ∈ ℤ∣n ≤ x} and we use the symbol ⌈x⌉ for rounding upwards.