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ETH Series in Information Theory and its Applications, Vol. 12
edited by Amos Lapidoth

Yiming Yan

Information-Theoretic

Aspects of Channel

State Quantization

1^st edition 2024. XIV, 164 pages, € 64,00.

ISBN 978-3-86628-833-1

Abstract

This thesis investigates channels with an altruistic helper, a party that observes the channel noise—or more generally, the channel state—and produces rate-limited quantization to assist the data transmission. Various notions of capacity, different revelation of assistance, and the impacts of whether the helper is aware of the message being transmitted or whether a feedback link exists are considered.

The thesis is structured in two primary parts. In the first part, four different notions of capacity are considered on the additive noise channels with a helper:

The first notion, referred to as the erasures-only capacity, requires that the decoder must avoid unconscious errors but may declare decoding failures with a small probability. We prove that on the memoryless modulo-additive noise channels (MMANCs), the erasures-only capacity matches the Shannon capacity. This result is generalized to continuous additive noise channels.

The second and third notions considered are the listsize capacity and the cutoff rate. The listsize capacity requires that the decoder generate a list containing all possible messages, and the ρ-th moment of the cardinality of that list converge to one for given ρ > 0. The cutoff rate is similar but restricts the list to messages at least as likely as the transmitted one. It is demonstrated that on the MMANCs, the listsize capacity equals the cutoff rate, and the same result is established on the Gaussian channel with decoder assistance.

The fourth notion examined is the zero-error capacity, which requires that the message be decoded with exactly zero probability of error. On the MMANCs, both the scenarios with and without feedback are studied.

In its presence, a complete solution of said capacity is provided. In its absence, a solution is provided when the alphabet size is prime. For all other cases, upper and lower bounds on the capacity are derived, leading to a necessary and sufficient condition for its positivity. Thanks to the helper, the zero-error capacity may increase by more than the helper’s rate, and it can be positive yet smaller than one bit.

The second part of the thesis focuses on the effects of message cognizance and feedback: In particular, the capacity of a state-dependent discrete memoryless channel (SD-DMC) is derived for the setting where a message-cognizant rate-limited helper observes the state sequence noncausally and provides its description to both encoder and decoder. Said capacity is not increased if a feedback link from the receiver to the encoder is introduced.

The same capacity is also derived for the Gaussian channel, and it is demonstrated that message cognizance increases the channel capacity. In this setting the feedback link—while not increasing capacity—eliminates the need for the helper’s cognition of the transmitted message. Moreover, in this setting, the results on capacity also hold for the cutoff rate and the listsize capacity.

Keywords: Additive-noise channel; cutoff rate; erasures-only capacity; feedback; helper; listsize capacity; state-dependent channel; zero-error capacity.

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