Two equivalent linear codes have the same weight enumerator
but the converse does not hold. We investigate which code is unique
up to equivalence in view of the weight enumerator. The main purpose
of this paper is to investigate the weight enumerators associated to
the only one equivalent class of linear codes and to construct these
codes. Furthermore, we construct non-equivalent linear codes with
the same weight enumerator and give the generator matrices of these
codes.