Multichannel Optimal Tree-Decodable Codes are Not Always Optimal Prefix Codes

TL;DR

This paper investigates multichannel prefix codes, revealing that optimal tree-decodable codes are not always optimal prefix codes, especially when more than two channels are involved, due to fundamental structural limitations.

Contribution

It provides a general sufficient condition for the existence of optimal tree-decodable codes that are not optimal prefix codes in multichannel settings.

Findings

Optimal tree-decodable codes are not always optimal prefix codes beyond two channels.

A fundamental structural reason causes non-tree-decodability in multichannel prefix codes.

A sufficient condition on channel alphabets determines when such codes exist.

Abstract

The theory of multichannel prefix codes aims to generalize the classical theory of prefix codes. Although single- and two-channel prefix codes always have decoding trees, the same cannot be said when there are more than two channels. One question is of theoretical interest: Do there exist optimal tree-decodable codes that are not optimal prefix codes? Existing literature, which focused on generalizing single-channel results, covered little about non-tree-decodable prefix codes since they have no single-channel counterparts. In this work, we study the fundamental reason behind the non-tree-decodability of prefix codes. By investigating the simplest non-tree-decodable structure, we obtain a general sufficient condition on the channel alphabets for the existence of optimal tree-decodable codes that are not optimal prefix codes.