Weighted Modules and Representations of Codes

A weight on a finite module is called egalitarian if the average weights of elements of any two its nonzero submodules are equal and it is called homogeneous if in addition the weights of any two associated elements are equal. The criteria of the existence of the egalitarian and homogeneous weights on an arbitrary finite module and the description of possible homogeneous weights are given. These results generalize the analogous results of Constantinescu and Heise for the ring $\mathbb Z_m$. Those finite modules which admit a homogeneous weight are called weighted and characterized in terms of the composition factors of their socle. A homogeneous weight in terms of Mцbius and Euler functions for finite modules is described and effectively calculated. As an application, besides the known presentation of the generalized Kerdock code, also isometric representations of the Golay codes and the generalized Reed–Muller codes as short linear codes over modules are given.