On decoding hyperbolic codes

Abstract

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.

Multivariate Goppa codes

“…where L ′ j (x j ) denotes the formal derivative of L j (x j ), defined in Equation (8). The polynomial L(x) plays an important role in determining the dual code C(S, A, h) ⊥ , which was studied in [21] in terms of the vanishing ideal of S and in [24] in terms of the indicator functions of S.…”

“…In [13], the authors provide a list decoding algorithm for the tensor product of GRS codes. In [8], they authors use the tensor product of GRS codes to decode hyperbolic codes, which are augmented Reed-Muller codes, in the sense that the dimension is greater than or equal, but the minimum distance is the same. The tensor product of GRS codes via a Goppa code is important in obtaining the following result, which is proved in Section 2.…”