[Paper Review] A new bound for the capacity of the deletion channel with high deletion probabilities

Marco Dalai|arXiv (Cornell University)|Apr 2, 2010

Wireless Communication Security Techniques4 references4 citations

TL;DR

This paper establishes that the limit of the binary deletion channel capacity normalized by (1−d) as d→1 exists and equals the infimum of C(d)/(1−d) over d∈(0,1). Using known bounds on C(d), this result yields the improved upper bound lim_{d→1} C(d)/(1−d) ≤ 0.4143, supporting the conjecture that C(d) may be convex on [0,1].

ABSTRACT

Let $C(d)$ be the capacity of the binary deletion channel with deletion probability $d$. It was proved by Drinea and Mitzenmacher that, for all $d$, $C(d)/(1-d)\geq 0.1185 $. Fertonani and Duman recently showed that $\limsup_{d o 1}C(d)/(1-d)\leq 0.49$. In this paper, it is proved that $\lim_{d o 1}C(d)/(1-d)$ exists and is equal to $\inf_{d}C(d)/(1-d)$. This result suggests the conjecture that the curve $C(d)$ my be convex in the interval $d\in [0,1]$. Furthermore, using currently known bounds for $C(d)$, it leads to the upper bound $\lim_{d o 1}C(d)/(1-d)\leq 0.4143$.

Motivation & Objective

• To resolve the open question of whether lim_{d→1} C(d)/(1−d) exists.
• To improve the upper bound on the asymptotic capacity of the binary deletion channel at high deletion probabilities.
• To investigate the potential convexity of the capacity function C(d) on [0,1] based on the behavior near d=1.
• To leverage known bounds on finite-blocklength capacity C_n(d) to derive tighter bounds on the high-deletion-capacity limit.

Proposed method

• Uses a channel model W_{n,k} that outputs a uniformly random k-bit subsequence from an n-bit input to approximate the deletion channel W^d.
• Applies Theorem 1 to show that C_n(d) approximates C(1−k/n) for large n, linking finite-blocklength and asymptotic capacities.
• Relies on Lemma 8 from Fertonani and Duman (2012) which bounds limsup_{d→1} C(d)/(1−d) ≤ (nC_{n,k}+1)/(k+1) for any n,k.
• Proves Theorem 2 by showing that the limsup bound from Lemma 8 converges to C(d')/(1−d') as d'→1, implying the limit exists and equals the infimum of C(d)/(1−d).
• Uses numerical evaluation of C_{17}(0.65)=0.145 from prior work to compute the improved upper bound 0.4143 via Corollary 1.
• Analyzes the behavior of C_n(d)/(1−d) as d→1, showing it tends to 1, which implies non-convexity for finite n, contrasting with the potential asymptotic convexity of C(d).

Experimental results

Research questions

• RQ1 Does the limit lim_{d→1} C(d)/(1−d) exist for the binary deletion channel?
• RQ2 Can tighter upper bounds on lim_{d→1} C(d)/(1−d) be derived using known finite-blocklength capacity bounds?
• RQ3 Is the capacity function C(d) convex on the interval [0,1], particularly near d=1?
• RQ4 How does the behavior of finite-blocklength capacity C_n(d) near d=1 compare to the asymptotic capacity C(d)?

Key findings

• The limit lim_{d→1} C(d)/(1−d) exists and is equal to inf_{d∈(0,1)} C(d)/(1−d), resolving an open question about the existence of this limit.
• The improved upper bound lim_{d→1} C(d)/(1−d) ≤ 0.4143 is established using the known bound C_{17}(0.65) = 0.145.
• The result suggests that C(d) may be convex on [0,1], as the limit being equal to the infimum is a necessary condition for convexity near d=1.
• For finite n, C_n(d)/(1−d) → 1 as d→1, indicating that C_n(d) is not convex near d=1, but this behavior does not preclude convexity of the limit function C(d).
• The existence of the limit allows the use of bounds on C(d) at moderate d (e.g., d=0.65) to derive provable bounds on the high-deletion-capacity limit.
• The paper provides a framework to improve the upper bound on lim_{d→1} C(d)/(1−d) by numerically evaluating C_n(d) for larger n near d≈0.65, though this is computationally intensive.