Codes as Elements in a Group Algebra

Abstract

Let F be a finite field with q elements, where q is a prime power, and let V:=Fn be the vector space of all n-tuples over F. A q-ary code of length n is a subset ℂ of V containing at least two elements. The vector space V has, with respect to vector addition, the structure of an elementary Abelian group. We shall describe the group algebra ℂ (V) of this group over the complex field ℂ as follows. To each element u∈V we associate a basis element E(u) of ℂ(V). The product of the basis elements E(u), E(v) is defined by $$ E\left( u \right)\,E\left( v \right) = E\left( {u + v} \right) $$ (1) so that multiplication by a given E(u) induces a permutation on the q basis elements. It is easily seen that these permutations correspond to the regular representation of the additive group of V. Thus, x2102(V) is the q -dimensional linear associative and commutative algebra over x2102 afforded by the regular representation of this group. Every subset C⊆V is represented by the element $$ C: = \sum\limits_{{u \in C}} {E\left( u \right)} $$ (2) in ℂ(V), for which we use the same symbol. In this way, a code C can be viewed as an element of x2102(V). The aim of this paper is to show the usefulness of this representation in some classical problems of coding theory. In section 2, we shall study the MacWilliams identities in connection with the Fourier transform.