  The determination of the weight distribution of linear codes has been a
fascinating problem since the very beginning of coding theory. There has been a
lot of research on weight enumerators of special cases, such as self-dual codes
and codes with small Singleton's defect. We propose a new set of linear
relations that must be satisfied by the coefficients of the weight
distribution. From these relations we are able to derive known identities (in
an easier way) for interesting cases, such as extremal codes, Hermitian codes,
MDS and NMDS codes. Moreover, we are able to present for the first time the
weight distribution of AMDS codes. We also discuss the link between our results
and the Pless equations.
