Refinements and Generalizations of the Shannon Lower Bound via Extensions of the Kraft Inequality

TL;DR

This paper improves the Shannon lower bound in data compression by extending the Kraft inequality for various coding scenarios.

Contribution

New extended versions of the Kraft inequality and refined Shannon lower bounds for different rate-distortion coding cases.

Findings

Sharper bounds for one-to-one and D-semifaithful codes are derived.

A Shannon lower bound for sliding-window distortion measures is established.

An individual-sequence version of the Shannon lower bound is introduced.

Abstract

We derive a few extended versions of the Kraft inequality for lossy compression, which pave the way to the derivation of several refinements and extensions of the well-known Shannon lower bound in a variety of instances of rate-distortion coding. These refinements and extensions include sharper bounds for one-to-one codes and D-semifaithful codes, a Shannon lower bound for distortion measures based on sliding-window functions, and an individual-sequence counterpart of the Shannon lower bound.