Nonuniform Fuchsian codes for noisy channels

Abstract

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm.Peer ReviewedPostprint (author’s final draft

Fuchsian codes with arbitrarily high code rates

“…Next we review some algebraic concepts and results related to Fuchsian groups acting on the complex upper half-plane, in particular those arising from quaternion algebras over a number field F , thus extending our work in [2].…”

“…In [2], we described in detail how to construct and decode Fuchsian codes in an AWGN channel. In order to make this paper self-contained, we shortly restate the process in this more general setting, focusing on some essential properties.…”

“…Actually, for a fixed Fuchsian group Γ the complexity of the decoding algorithm depends only on the selection of the subset S Γ . The choice of the center of the code τ will influence the performance of the code, being related to the minimum border distance, as stated in [2] (see the code design criterion therein). In the case F = Q, when we restrict to the natural order Z[1, I, J, K], a complex number γ(τ ) to be transmitted is identified with the matrix γ ∈ S Γ ⊂ φ(O * + ) (recall that we take τ an interior point of the fundamental domain).…”

“…Apart from the sub-linear decoding complexity, another advantage of the Fuchsian codes in [2] is that they allow, in the sense of the above definition, to compress information. Namely, they enable us to embed three independent integers in one complex number to be transmitted, having thus code rate R = 3.…”

“…Nevertheless, our decoder will rely on hyperbolic geometry as opposed to the classical decoders based on Euclidean geometry. The paper is organized as follows: in Section 2 we give some background and notation on Fuchsian groups acting on the complex upper half-plane, specially those coming from quaternion algebras over a number field F , in a more general setting than [4] and [2]. In Section 3, we generalize the construction of Fuchsian codes in order to obtain codes of arbitrarily high rates.…”