One-Shot Capacity of Discrete Channels

TL;DR

This paper characterizes the maximum amount of information that can be transmitted in a single use of a discrete channel with a specified error probability, addressing scenarios where block length is limited or one-shot transmission is needed.

Contribution

It provides a precise characterization of one-shot capacity for discrete channels, a significant departure from traditional asymptotic capacity definitions.

Findings

Derived a formula for one-shot capacity with error constraints

Showed the difference between one-shot and zero-error capacities

Applicable to scenarios with limited or single-use transmissions

Abstract

Shannon defined channel capacity as the highest rate at which there exists a sequence of codes of block length $n$ such that the error probability goes to zero as $n$ goes to infinity. In this definition, it is implicit that the block length, which can be viewed as the number of available channel uses, is unlimited. This is not the case when the transmission power must be concentrated on a single transmission, most notably in military scenarios with adversarial conditions or delay-tolerant networks with random short encounters. A natural question arises: how much information can we transmit in a single use of the channel? We give a precise characterization of the one-shot capacity of discrete channels, defined as the maximum number of bits that can be transmitted in a single use of a channel with an error probability that does not exceed a prescribed value. This capacity definition is shown to be useful and significantly different from the zero-error problem statement.