Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Quantitative Version of More Capable Channel Comparison

Donald Kougang-Yombi, Jan Hązła

TL;DR

This work generalizes the classical 'more capable' ordering to a quantitative 'more capable with advantage' relation, $W+\eta\succeq_{mc}\widetilde{W}$, and proves tensorization for product channels. It provides explicit characterizations for the $\mathrm{BEC}$ and $\mathrm{BSC}$ pair, enabling cross-channel bounds that transfer capacity-achieving properties from the $BEC$ to the $BSC$, and vice versa. Two main applications are developed: (i) a novel list-decoding bound for transitive linear codes achieving capacity on the $BEC$, with list size $L\le 2^{(\eta+\varepsilon)n}$, and (ii) improved lower bounds on the entropy rate of binary hidden Markov processes via the advantaged channel framework. Together, these results offer tighter cross-channel insights for coding and stochastic process analysis, and highlight the utility of quantitative channel orderings in deriving performance guarantees.

Abstract

This paper introduces a quantitative generalization of the ``more capable'' comparison of broadcast channels, which is termed ``more capable with advantage''. Some basic properties are demonstrated (including tensorization on product channels), and a characterisation is given for the cases of Binary Symmetric Channel (BSC) and Binary Erasure Channel (BEC). It is then applied to two problems. First, a list decoding bound on the BSC is given that applies to transitive codes that achieve capacity on the BEC. Second, new lower bounds on entropy rates of binary hidden Markov processes are derived.

A Quantitative Version of More Capable Channel Comparison

TL;DR

This work generalizes the classical 'more capable' ordering to a quantitative 'more capable with advantage' relation, , and proves tensorization for product channels. It provides explicit characterizations for the and pair, enabling cross-channel bounds that transfer capacity-achieving properties from the to the , and vice versa. Two main applications are developed: (i) a novel list-decoding bound for transitive linear codes achieving capacity on the , with list size , and (ii) improved lower bounds on the entropy rate of binary hidden Markov processes via the advantaged channel framework. Together, these results offer tighter cross-channel insights for coding and stochastic process analysis, and highlight the utility of quantitative channel orderings in deriving performance guarantees.

Abstract

This paper introduces a quantitative generalization of the ``more capable'' comparison of broadcast channels, which is termed ``more capable with advantage''. Some basic properties are demonstrated (including tensorization on product channels), and a characterisation is given for the cases of Binary Symmetric Channel (BSC) and Binary Erasure Channel (BEC). It is then applied to two problems. First, a list decoding bound on the BSC is given that applies to transitive codes that achieve capacity on the BEC. Second, new lower bounds on entropy rates of binary hidden Markov processes are derived.
Paper Structure (19 sections, 19 theorems, 53 equations, 4 figures)

This paper contains 19 sections, 19 theorems, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. List decoding
  3. Hidden Markov process entropy rate
  4. Organization
  5. Definitions and basic properties
  6. Characterization for the $\mathop{\mathrm{BEC}}\nolimits_q$ and the $\mathop{\mathrm{BSC}}\nolimits_p$
  7. More capable with advantage
  8. Less noisy with advantage
  9. List decoding
  10. Proof of \ref{['thm4.2']}
  11. Proof of \ref{['Thm1']}
  12. Hidden Markov processes
  13. Acknowledgments
  14. Appendix
  15. Proof of \ref{['mc-tensorize']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $0\le p,q\le 1$ such that $q\le 4p(1-p)$. Then, $\mathop{\mathrm{BSC}}\nolimits_p+h(p)-q\succeq_{\mathrm{mc}}\mathop{\mathrm{BEC}}\nolimits_q$.

Figures (4)

  • Figure 1: Illustration of our list decoding bounds.
  • Figure 2: Comparison of our lower bounds for $\overline{H}(Y)$, that given in ordentlich2016novel, and the true rate $\overline{H}(Y)$ as computed by an approximation algorithm.
  • Figure 3: Shapes of $f$ and its concave (in blue) and convex (in green) envelopes for $q=0.5$ and different values of $p$.
  • Figure 4: Comparison of $\gamma$ from ordentlich2016novel (i.e. $\gamma=4\alpha(1-\alpha)$) and that of \ref{['pro5.3']} which is the argmax of the right-hand side of \ref{['eq1.4']}.

Theorems & Definitions (48)

  • Theorem 1.1: Corollary 1.9 in samorodnitsky2016entropy
  • Theorem 1.2
  • Theorem 1.3: Theorem 2 in ordentlich2016novel
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Claim 2.6
  • ...and 38 more