Lossless Data Compression with Error Detection using Cantor Set

In 2009, a lossless compression algorithm based on 1D chaotic maps known as Generalized Lur\"{o}th Series (or GLS) has been proposed. This algorithm (GLS-coding) encodes the input message as a symbolic sequence on an appropriate 1D chaotic map (GLS) and the compressed file is obtained as the initial value by iterating backwards on the map. For ergodic sources, it was shown that GLS-coding achieves the best possible lossless compression (in the noiseless setting) bounded by Shannon entropy. However, in the presence of noise, even small errors in the compressed file leads to catastrophic decoding errors owing to sensitive dependence on initial values. In this paper, we first show that Repetition codes $\mathcal{R}_n$ (every symbol is repeated $n$ times, where $n$ is a positive odd integer), the oldest and the most basic error correction and detection codes in literature, actually lie on a Cantor set with a fractal dimension of $\frac{1}{n}$, which is also the rate of the code. Inspired by this, we incorporate error detection capability to GLS-coding by ensuring that the compressed file (initial value on the map) lies on a Cantor set of measure zero. Even a 1-bit error in the initial value will throw it outside the Cantor set which can be detected while decoding. The error detection performance (and also the rate of the code) can be controlled by the fractal dimension of the Cantor set and could be suitably adjusted depending on the noise level of the communication channel.