Multidimensional Fibonacci Coding

Abstract

Fibonacci codes are self-synchronizing variable-length codes that are proven useful for their robustness and compression capability. Asymptotically, these codes provide better compression efficiency as the order of the underlying Fibonacci sequence increases but at the price of the increased suffix length. We propose a circumvention to this problem by introducing higher-dimensional Fibonacci codes for integer vectors. The resulting multidimensional Fibonacci coding is comparable to the classical one in terms of compression; while encoding several numbers all at once for a shared suffix generally results in a shorter codeword, the efficiency takes a backlash when terms from different orders of magnitude are encoded together. In addition, while laying the groundwork for the new encoding scheme, we provide extensive theoretical background and generalize the theorem of Zeckendorf to higher order. As such, our work unifies several variations of Zeckendorf’s theorem while also providing new grounds for its legitimacy.

Keywords

• Fibonacci code
• Zeckendorf’s theorem
• prefix code
• data compression
• ℤ-module
• Gaussian integers