On an analogue of the doubling method in coding theory
It is well known that there is a deep relationship between codes and lattices. Concepts from coding theory are related to concepts of lattice theory as, for example, weight enumerators to theta series, MacWilliams identity to Jacobi identity, and Gleason's theorem to Hecke's theorem. In this framework, higher-genus (or multiple) weight enumerators are related to Siegel theta series, which opens up the possibility of introducing concepts from the theory of higher-rank modular forms to coding theory. There has been important work in this direction, for example Runge introduced a coding theory analogue of Siegel's $Φ$-operator and Nebe analogues of Hecke operators. In this paper, we show that the celebrated Doubling Method from the theory of higher-rank modular forms has a coding theory analogue. Given the impact that the Doubling Method has had in the study of higher-rank modular forms, one may expect that its analogue may prove useful to the study of higher-genus weight enumerators. In this paper we use it to solve an analogue of the "basis problem". That is, we express "cuspidal" polynomials which are invariant under a Clifford-Weil type group as an explicit linear combination of higher-genus weight enumerators of self-dual codes of that type.