Metric structure of linear codes

The bilinear form with associated identity matrix is used in coding theory to define the dual code of a linear code, furthermore it endows linear codes with a metric space structure. We used this study of bilinear forms over a finite field to give a decomposition of an arbitrary linear code similar to the one obtained for generalized toric codes. Such a decomposition, called geometric decomposition of a linear code, can be obtained in a constructive way; it allows us to express easily the dual code of a linear code and provides us with a method to estimate their minimum distance.