Permutation codes, source coding and a generalisation of Bollob\'as-Lubell-Yamamoto-Meshalkin and Kraft inequalities

TL;DR

This paper introduces a broad framework for establishing Kraft-type inequalities for prefix-free permutation codes in source coding, revealing limitations of McMillan's converse and extending classical results to more general structures.

Contribution

It generalizes Kraft inequalities to permutation codes and other structures, providing new counterexamples to McMillan's converse theorem and unifying classical results as corollaries.

Findings

Kraft-type inequalities are proven for various permutation code notions.

Counterexamples show McMillan's converse does not always hold.

Classical Kraft inequality is a special case of the developed framework.

Abstract

We develop a general framework to prove Kraft-type inequalities for prefix-free permutation codes for source coding with various notions of permutation code and prefix. We also show that the McMillan-type converse theorem in most of these cases does not hold, and give a general form of a counterexample. Our approach is more general and works for other structures besides permutation codes. The classical Kraft inequality for prefix-free codes as well as results about permutation codes follow as corollaries of our main theorem and main counterexample.