19 Self-dual codes and invariant theory

Abstract

A formal notion of a Typ T of a self-dual linear code over a nite left R- module V is introduced which allows to give explicit generators of a nite complex matrix group, the associated Clifford-Weil group C.T / • GLjVj.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/ D 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 / D C.T m /. The nite Siegel 8-operator mapping cwem.C/ to cwemi1.C/ hence denes a ring epimorphism 8m V Inv.Cm.T // ! Inv.Cmi1.T // between invariant rings of complex matrix groups of different degrees. If R D V is a - nite eld, then the structure of Cm.T / allows to dene 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 CV N.