# On Universal Codes for Integers: Wallace Tree, Elias Omega and Variations

## TL;DR

This paper introduces and analyzes the Wallace Tree Code (WTC), a universal integer coding method based on binary trees, comparing it to Fibonacci and Elias omega codes, with proposed improvements to Elias omega.

## Contribution

It presents the encoding and decoding routines for WTC, analyzes its properties, and proposes enhancements to Elias omega code, advancing universal integer coding techniques.

## Key findings

- WTC offers competitive compression performance
- WTC's properties are comparable to Fibonacci and Elias omega codes
- Proposed improvements enhance Elias omega code efficiency

## Abstract

A universal code for the (positive) integers can be used to store or compress a sequence of integers. Every universal code implies a probability distribution on integers. This implied distribution may be a reasonable choice when the true distribution of a source of integers is unknown. Wallace Tree Code (WTC) is a universal code for integers based on binary trees. We give the encoding and decoding routines for WTC and analyse the properties of the code in comparison to two well-known codes, the Fibonacci and Elias omega codes. Some improvements on the Elias omega code are also described and examined.