MacWilliams Identites for Linear Codes with Group Action(Algebraic Combinatorial Theory)
Tomoyuki Yoshida · Institutional Repositories DataBase (IRDB) · 1988
Let $Set^{G}$ denote the category of G-sets and G-maps; and let $Set_{f}^{G}$ denote the category of finite G-sets.If we consider combinatorial theory as a theory of Set $f$ , the category of finite sets, then the theory of $Set_{f}^{G}$ can be consider as equivariant combinatorial theory.I have studied Fisher's inequality for block designs with group action based on this idea([Yo87]).Fortunately, this idea can be also applied to the formulation and the proof of the MacWilliams identity for linear codes with group action.Because the category of G-sets has similar properties as the category of sets, we can formally extend usual theories to theories with group action: that is, equivariant versions of theories.However the category of G-sets has many non-isomorphic connected objects (transitive G-sets) in addition to the terminal object, if $G$ is not trivial.For this reason, it is often unavoid- able that such theories become too difficult to study them directly.For example, the theory of vector spaces is easy, but the theory of vector spaces with group action is nothing but the representation theory of groups.Even the definition of the concept of matrices is not trivial in this theory.