#### On Optimal Latency of Communications

##### Minh Au, Francois Gagnon

In this paper we investigate the optimal latency of communications. Focusing on fixed rate communication without any feedback channel, this paper encompasses low-latency strategies with which one hop and multi-hop communication issues are treated from an information theoretic perspective. By defining the latency as the time required to make decisions, we prove that if short messages can be transmitted in parallel Gaussian channels, for example, via orthogonal frequency-division multiplexing (OFDM)-like signals, there exists an optimal low-latency strategy for every code. This can be achieved via early-detection schemes or asynchronous detections. We first provide the optimal achievable latency in additive white Gaussian noise (AWGN) channels for every channel code given a probability block error $\epsilon$. This can be obtained via sequential ratio tests or a "genie" aided, \textit{e.g}. error-detecting codes. Results demonstrate the effectiveness of the approach. Next, we show how early-detection can be effective with OFDM signals while maintaining its spectral efficiency via random coding or pre-coding random matrices. Finally, we explore the optimal low-latency strategy in multi-hop relaying schemes. For amplify-and-forward (AF) and decode-and-forward (DF) relaying schemes there exist an optimal achievable latency. In particular, we first show that there exist a better low-latency strategy, for which AF relays could transmit while receiving. This can be achieved by using amplify and forward combined with early detection.

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