A Linear-Size Block-Partition Fibonacci Encoding for G\"odel Numbering

TL;DR

This paper introduces a novel Fibonacci-based encoding for finite strings that is size-efficient, injective, and guarantees unique decoding, outperforming previous exponential growth methods.

Contribution

The paper presents a new block-partition Fibonacci encoding that achieves linear-size growth and unique representation for string encoding, improving over prior exponential methods.

Findings

Encoding size grows as Θ(m) for strings of length m.

The encoding is injective and allows string length recovery.

It avoids exponential digit growth seen in previous methods.

Abstract

We construct an encoding of finite strings over a fixed finite alphabet as natural numbers, based on a block partition of the Fibonacci sequence. Each position in the string selects one Fibonacci number from a dedicated block, with unused indices between blocks guaranteeing non-adjacency. The encoded number is the sum of the selected Fibonacci numbers, and Zeckendorf's theorem guarantees that this sum uniquely determines the selection. The encoding is injective, the string length is recoverable from the code, and the worst-case digit count of the encoded number grows as $\Theta(m)$ for strings of length $m$, matching the information-theoretic lower bound up to a constant factor. We also prove that the natural right-nested use of Rosko's (2025) binary carryless pairing for sequence encoding has worst-case $\Theta(2^m)$ digit growth, an exponential blowup that the block-partition construction avoids entirely.