Equivariant theory for codes and lattices I

TL;DR

This paper generalizes key theorems connecting codes and lattices over finite Frobenius rings, extending MacWilliams identities and exploring the relationship between codes and lattices.

Contribution

It introduces a broad generalization of Hayden's theorem and Astumi's MacWilliams identity for $G$-codes over finite Frobenius rings, including lattice analogues.

Findings

Generalized Hayden's theorem for $G$-codes

Extended MacWilliams identities for various weight enumerators

Established links between $G$-codes and $G$-lattices

Abstract

In this paper, we present a generalization of Hayden's theorem [7, Theorem 4.2] for $G$-codes over finite Frobenius rings. A lattice theoretical form of this generalization is also given. Moreover, Astumi's MacWilliams identity [1, Theorem 1] is generalized in several ways for different weight enumerators of $G$-codes over finite Frobenius rings. Furthermore, we provide the Jacobi analogue of Astumi's MacWilliams identity for $G$-codes over finite Frobenius rings. Finally, we study the relation between $G$-codes and its corresponding $G$-lattices.