Variable-Length Coding with Cost Allowing Non-Vanishing Error Probability

TL;DR

This paper derives a general formula for the minimum achievable rate in fixed-to-variable length coding with a regular cost function, allowing a non-zero error probability, and explores the relationship between different cost functions.

Contribution

It introduces a unified formula for the minimum achievable rate with non-vanishing error probability and reveals the equivalence of dominant source sets across different cost functions.

Findings

Derived a general formula for the minimum achievable rate with error tolerance.

Showed the dominant set of source sequences is consistent across different cost functions.

Provided a formula for the second-order minimum achievable rate.

Abstract

We derive a general formula of the minimum achievable rate for fixed-to-variable length coding with a regular cost function by allowing the error probability up to a constant $\varepsilon$. For a fixed-to-variable length code, we call the set of source sequences that can be decoded without error the dominant set of source sequences. For any two regular cost functions, it is revealed that the dominant set of source sequences for a code attaining the minimum achievable rate with a cost function is also the dominant set for a code attaining the minimum achievable rate with the other cost function. We also give a general formula of the second-order minimum achievable rate.