# Optimal Age over Erasure Channels

## TL;DR

This paper investigates optimal coding strategies to minimize the average age of information over erasure channels, providing closed-form solutions for equal alphabet sizes and bounds for different sizes, advancing understanding of age optimization in communication systems.

## Contribution

It introduces a novel analysis of age minimization over erasure channels, deriving closed-form solutions for equal alphabet sizes and bounds for differing sizes, using random coding arguments.

## Key findings

- Trivial coding strategy is optimal when source and channel alphabets are equal.
- Closed-form expression for average age in the equal alphabet case.
- Random coding approaches approach optimal age as source alphabet size increases.

## Abstract

Previous works on age of information and erasure channels have dealt with specific models and computed the average age or average peak age for certain settings. In this paper, given a source that produces a letter every $T_s$ seconds and an erasure channel that can be used every $T_c$ seconds, we ask what is the coding strategy that minimizes the time-average age of information that an observer of the channel output incurs. We first analyze the case where the source alphabet and the channel-input alphabet have the same size. We show that a trivial coding strategy is optimal and a closed form expression for the age can be derived. We then analyze the case where the alphabets have different sizes. We use a random coding argument to bound the average age and show that the average age achieved using random codes converges to the optimal average age of linear block codes as the source alphabet becomes large.