Fuchsian codes with arbitrarily high code rate

TL;DR

This paper introduces Fuchsian codes with arbitrarily high code rates, derived from quaternion algebras, maintaining logarithmic decoding complexity, thus enhancing data transmission efficiency over noisy channels.

Contribution

It constructs Fuchsian codes with arbitrarily high code rates based on quaternion algebra groups, improving upon previous high-rate codes while preserving efficient decoding.

Findings

Achieved code rate proportional to the degree of the base field

Constructed explicit codes with code rate six

Maintained logarithmic decoding complexity

Abstract

Recently, so-called Fuchsian codes have been proposed in [I. Blanco-Chac\'on et al., "Nonuniform Fuchsian codes for noisy channels", J. of the Franklin Institute 2014] for communication over channels subject to additive white Gaussian noise (AWGN). The two main advantages of Fuchsian codes are their ability to compress information, i.e., high code rate, and their logarithmic decoding complexity. In this paper, we improve the first property further by constructing Fuchsian codes with arbitrarily high code rates while maintaining logarithmic decoding complexity. Namely, in the case of Fuchsian groups derived from quaternion algebras over totally real fields we obtain a code rate that is proportional to the degree of the base field. In particular, we consider arithmetic Fuchsian groups of signature (1;e) to construct explicit codes having code rate six, meaning that we can transmit six independent integers during one channel use.