# Decision Procedure for the Existence of Two-Channel Prefix-Free Codes

## TL;DR

This paper presents an efficient decision procedure for determining the existence of two-channel prefix-free codes by transforming the problem into a constrained rectangle packing problem, which can be solved efficiently despite its NP-completeness in general.

## Contribution

It introduces a novel algorithm that efficiently decides the existence of two-channel prefix-free codes by leveraging a constrained rectangle packing formulation.

## Key findings

- The algorithm effectively solves the two-channel prefix-free code existence problem.
- The approach relates coding problems to geometric packing problems.
- It demonstrates that additional constraints enable efficient solutions despite NP-completeness in general.

## Abstract

The Kraft inequality gives a necessary and sufficient condition for the existence of a single channel prefix-free code. However, the multichannel Kraft inequality does not imply the existence of a multichannel prefix-free code in general. It is natural to ask whatever there exists an efficient decision procedure for the existence of multichannel prefix-free codes. In this paper, we tackle the two-channel case of the above problem by relating it to a constrained rectangle packing problem. Although a general rectangle packing problem is NP-complete, the extra imposed constraints allow us to propose an algorithm which can solve the problem efficiently.