# Exponent Function for Stationary Memoryless Channels with Input Cost at Rates above the Capacity

## TL;DR

This paper investigates the decay rate of correct decoding probability for stationary memoryless channels with input cost at rates exceeding capacity, introducing a recursive method to determine the optimal exponent function.

## Contribution

It introduces a recursive technique based on the information spectrum approach to analyze the exponential decay of decoding correctness above capacity.

## Key findings

- Correct decoding probability tends to zero exponentially above capacity.
- Determined the optimal exponent function for finite input/output sets.
- Developed a new recursive method for analyzing information spectrum quantities.

## Abstract

We consider the stationaly memoryless channels with input cost. We prove that for transmission rates above the capacity the correct probability of decoding tends to zero exponentially as the block length $n$ of codes tends to infinity. In the case where both of channel input and output sets are finite, we determine the optimal exponent function on the above exponential decay of the correct probability. To derive this result we use a new technique called the recuresive method, which is based on the information spectrum approach. The recursive method utilize a certain recursive structure on the information spectrum quantities.