Self-dual codes and invariant theory
Nebe Gabriele · NATO science for peace and security series. D, Information and communication security · 2009
A formal notion of a Typ T of a self-dual linear code over a finite left R-module 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 ≤ (Vm)Nof Type Tmwith associated Clifford-Weil group 𝒞m(T) = 𝒞(Tm). The finite Siegel &PHgr;-operator mapping cwem(C) to cwem−1(C) hence defines a ring epimorphism &PHgr;m: 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 &PHgr;m. The Hecke-eigenspaces yield explicit linear relations among the cwemof self-dual codes C ≤ VN.