Self-dual codes and invariant theory
نویسنده
چکیده
Abstract. A formal notion of a Typ T of a self-dual linear code over a finite left Rmodule V is introduced which allows to give explicit generators of a finite complex matrix group, the associated Clifford-Weil group C(T ) ≤ GL|V |(C), such that the complete weight enumerators of self-dual isotropic codes of Type T span the ring of invariants of C(T ). This generalizes Gleason’s 1970 theorem to a very wide class of rings and also includes multiple weight enumerators (see Section 2.7), as these are the complete weight enumerators cwem (C) = cwe(Rm ⊗ C) of Rm×m -linear self-dual codes Rm⊗C ≤ (V m )N of Type T m with associated Clifford-Weil group Cm (T ) = C(T m ). The finite Siegel 8-operator mapping cwem (C) to cwem−1(C) hence defines a ring epimorphism 8m : Inv(Cm (T )) → Inv(Cm−1(T )) between invariant rings of complex matrix groups of different degrees. If R = V is a finite field, then the structure of Cm (T ) allows to define a commutative algebra of Cm (T ) double cosets, called a Hecke algebra in analogy to the one in the theory of lattices and modular forms. This algebra consists of self-adjoint linear operators on Inv(Cm (T )) commuting with 8m . The Hecke-eigenspaces yield explicit linear relations among the cwem of self-dual codes C ≤ V N .
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