On decoding hyperbolic codes
Eduardo Camps-Moreno, Ignacio García-Marco, Hiram H. López, Irene Márquez-Corbella, Edgar Martínez-Moro, Eliseo Sarmiento
TL;DR
The paper tackles decoding hyperbolic codes $\mathrm{Hyp}_q(d,m)$ by leveraging their containment within and containment of Reed-Muller codes $\mathrm{RM}_q(s,m)$, enabling decoding via known RM decoders or via decoders for containing RM codes. It develops strategies based on the closest large contained RM code or the closest small containing RM code, and discusses a combined approach for $m=2$. The work further links hyperbolic codes to Cube codes (tensor products of Reed-Solomon codes) and adapts Geil and Matsumoto's generalized Sudan list decoding to order-domain codes, with detailed constructions and complexity considerations. Overall, it provides a suite of decoding paradigms, analyzes their trade-offs, and demonstrates how to translate decoding tasks from hyperbolic codes to more tractable RM or Cube-code decoders, guiding practical decoder selection and future improvements.
Abstract
This work studies several decoding algorithms for hyperbolic codes. We use some previous ideas to describe how to decode a hyperbolic code using the largest Reed-Muller code contained in it or using the smallest Reed-Muller code that contains it. A combination of these two algorithms is proposed 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 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.