On McMillan's theorem about uniquely decipherable codes

TL;DR

This paper revisits McMillan's theorem on uniquely decipherable codes, presenting a polynomial-based proof that strengthens the original theorem by linking Kraft sums to non-commuting indeterminate evaluations.

Contribution

It introduces a polynomial approach involving non-commuting indeterminates to provide a strengthened proof of McMillan's theorem.

Findings

Polynomial method confirms Kraft sum bounds for uniquely decipherable codes

Strengthened version of McMillan's theorem established

New proof technique involving non-commuting indeterminates

Abstract

Karush's proof of McMillan's theorem is recast as an argument involving polynomials with non-commuting indeterminates certain evaluations of which yield the Kraft sums of codes, proving a strengthened version of McMillan's theorem.