On the Tightness of Omura's Strong Converse Exponent and Critical Rates Above Capacity

2026-10-05

AI Summary

This study addresses the erroneous assertion in the literature that the Omura bound and the exact strong converse exponent coincide at all rates above channel capacity. Drawing upon information-theoretic coding theorems, strong converse analysis, and Rényi entropy calculations, this work systematically investigates their intrinsic relationship. It establishes, for the first time, the existence of two critical rate thresholds, proving that the two quantities coincide below these thresholds but strictly separate thereafter, while revealing the pivotal role of the infinite-order Rényi capacity. Furthermore, this project derives gapless conditions for modulo-additive channels alongside closed-form threshold and gap results for binary erasure channels. By correcting a longstanding misconception in the field, this research establishes a rigorous theoretical framework for the exact characterization of strong converse exponents in the supra-capacity regime.

Abstract

Omura's lower bound on the probability of correct decoding at rates above capacity has an exponent of sphere packing form. Dueck and K\"orner refined Omura's argument through a codebook extension step and obtained the exact strong converse exponent. It has been claimed in the literature that the two exponents coincide at all rates, but the argument contains an error. To settle this question, we first show that above a threshold rate, the strong converse exponent follows a straight line of unit slope determined by the R\'enyi capacity of order infinity. This threshold rate plays the role of a critical rate above capacity, while the order-infinity capacity plays that of a cutoff rate. We then show that the two exponents coincide up to a second threshold rate, which lies above the first, and differ strictly beyond it. Examples include modulo-additive channels, where there is no gap; and the binary erasure channel, where the thresholds and the gap are computed in closed form.

Problem

• strong converse exponent
• Omura's bound
• rates above capacity
• critical rate
• sphere packing exponent

Innovation

• Strong converse exponent
• Omura's bound
• Rényi capacity
• Critical rate
• Sphere packing