Vector spaces over finite fields

Prototypes of many combinatorial designs come from finite projective geometries and finite affine geometries . Vector spaces over finite fields provide a natural setting for describing these geometries. Among the numerous incidence structures that can be constructed using affine and projective geometries are infinite families of symmetric designs, nets and Latin squares . Subspaces of a vector space over a finite field can be regarded as linear codes that will be used in later chapters for constructing other combinatorial structures, such as Witt designs and balanced generalized weighing matrices . Finite fields In this section we recall a few basic results on finite fields which will be used throughout this book. For any prime p , the residue classes modulo p with the usual addition and multiplication form a finite field GF ( p ) of order p . These fields are called prime fields . Any finite field F of characteristic p contains GF ( p ) as a subfield. The field F then can be regarded as a finite-dimensional vector space over GF ( p ), and therefore, | F | = p n where n is the dimension of this vector space. Conversely, for any prime power q = p n , there is a unique (up to isomorphism) finite field of order q . This field is denoted by GF ( q ) and is often called the Galois field of order q . In general, the field GF ( q ) is isomorphic to (a unique) subfield of the field GF ( r ) if and only if r is a power of q .