We define Galois coverings of tropical curves for which a Galois correspondence and a universal mapping property hold.

A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code [Formula: see text], we find the largest Reed–Muller code contained in [Formula: see text] and the smallest Reed–Muller code containing [Formula: see text]. We then prove that similar to Reed–Muller and affine Cartesian codes, the [Formula: see text]th generalized Hamming weight and the [Formula: see text]th footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the [Formula: see text]th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the [Formula: see text]th footprint of a hyperbolic code that, sometimes, are sharp.

In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in [Formula: see text], [Formula: see text]. Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control 55(11) (2009) 4890–4897] that, contrary to what happens for codes over [Formula: see text], where [Formula: see text] is a field, when dealing with [Formula: see text] there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called [Formula: see text]-encoder. In this paper we solve this conjecture for a class of [Formula: see text] convolutional codes over [Formula: see text] and show that, in fact, these codes always admit a non-catastrophic [Formula: see text]-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic [Formula: see text]-encoder.

We introduce continuity filters as a generalization of Gabriel filters, still with the possibility of defining modules of quotients, which we present and give some properties about them. We describe continuity filters on principal ideal domains, and finite products of rings. We also study continuity filters with a least element.