Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Using Pseudocodewords to Transmit Information

Nathan Axvig

TL;DR

A new modulation scheme is developed, termed insphere modulation, that is capable of reliably transmitting both codewords and pseudocodewords that has higher spectral efficiencies than the binary linear codes from which they are derived and in some cases have the same or slightly better block error rates.

Abstract

The linear programming decoder will occasionally output fractional-valued sequences that do not correspond to binary codewords - such outputs are termed nontrivial pseudocodewords. Feldman et al. have demonstrated that it is precisely the presence of nontrivial pseudocodewords that prevents the linear programming decoder from attaining maximum-likelihood performance. The purpose of this paper is to cast a positive light onto these nontrivial pseudocodewords by turning them into beneficial additions to our codebooks. Specifically, we develop a new modulation scheme, termed insphere modulation, that is capable of reliably transmitting both codewords and pseudocodewords. The resulting non-binary, non-linear codebooks have higher spectral efficiencies than the binary linear codes from which they are derived and in some cases have the same or slightly better block error rates. In deriving our new modulation scheme we present an algorithm capable of computing the insphere of a polyhedral cone - which loosely speaking is the largest sphere contained within the cone. This result may be of independent mathematical interest.

Using Pseudocodewords to Transmit Information

TL;DR

A new modulation scheme is developed, termed insphere modulation, that is capable of reliably transmitting both codewords and pseudocodewords that has higher spectral efficiencies than the binary linear codes from which they are derived and in some cases have the same or slightly better block error rates.

Abstract

The linear programming decoder will occasionally output fractional-valued sequences that do not correspond to binary codewords - such outputs are termed nontrivial pseudocodewords. Feldman et al. have demonstrated that it is precisely the presence of nontrivial pseudocodewords that prevents the linear programming decoder from attaining maximum-likelihood performance. The purpose of this paper is to cast a positive light onto these nontrivial pseudocodewords by turning them into beneficial additions to our codebooks. Specifically, we develop a new modulation scheme, termed insphere modulation, that is capable of reliably transmitting both codewords and pseudocodewords. The resulting non-binary, non-linear codebooks have higher spectral efficiencies than the binary linear codes from which they are derived and in some cases have the same or slightly better block error rates. In deriving our new modulation scheme we present an algorithm capable of computing the insphere of a polyhedral cone - which loosely speaking is the largest sphere contained within the cone. This result may be of independent mathematical interest.

Paper Structure

This paper contains 8 sections, 9 theorems, 9 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Background on Linear Programming
  4. Background on Linear Programming Decoding
  5. How To Transmit a Pseudocodeword
  6. A Method of Computing the Insphere of a Polyhedral Cone
  7. Extending The Modulation Scheme via $\mathcal{C}$-Symmetry
  8. A Detailed Example

Key Result

Proposition 3.1

Let $\mathcal{M} = \{{\mathbf{x}} \, | \, A{\mathbf{x}} \geq {\mathbf{b}} \}$ be a polyhedron in $\mathbb{R}^n$, let ${\mathbf{x}}^\ast \in \mathcal{M}$ be given, and define $S$ to be the set of all indices $i$ where ${\mathbf{a}}_i^T {\mathbf{x}}^\ast = b_i$ - i.e., the index set for all constraint

Figures (3)

  • Figure 1: An illustration of our iterative method for computing the insphere of a polyhedral cone to any degree of accuracy. For this small example in $\mathbb{R}^2$, the exact insphere is found in only two steps.
  • Figure 2: A plot of upper bounds $z_\ell$ and lower bounds $w_\ell$ on the inradius for an approximation cone used to compute a transmittable version of the pseudocodeword ${\boldsymbol{\omega}}_3$ (see Section \ref{['sec:example']}). The cone is described by 35 constraints in $\mathbb{R}^{16}$. In this example, it takes 56 iterations to ensure that the upper and lower bounds are within $10^{-5}$ of one another.
  • Figure :

Theorems & Definitions (18)

  • Definition 2.1: fwk05
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Definition 4.1: Henrion2010
  • Proposition 4.2
  • Lemma 4.3
  • ...and 8 more