On decoding hyperbolic codes

Few decoding algorithms for hyperbolic codes are known in the literature, this article tries to fill this gap. The first part of this work compares hyperbolic codes and Reed-Muller codes. In particular, we determine when a Reed-Muller code is a hyperbolic code. As a byproduct, we state when a hyperbolic code has greater dimension than a Reed-Muller code when they both have the same minimum distance. We use the previous ideas to describe how to decode a hyperbolic code using the largest Reed-Muller code contained in it, or alternatively using the smallest Reed-Muller code that contains it. A combination of these two algorithms is proposed for the case when hyperbolic codes are defined by polynomials in two variables. Then, we compare hyperbolic codes and Cube codes (tensor product of Reed-Solomon codes) and we propose decoding algorithms of hyperbolic codes based on their closest Cube codes. Finally, we adapt to hyperbolic codes the Geil and Matsumoto's generalization of Sudan's list decoding algorithm.