Zeckendorf's Theorem and Fibonacci Coding for Modules

Zeckendorf’s theorem states that every positive integer can be written uniquely as a sum of nonconsecutive Fibonacci numbers. This theorem induces a binary numeration system for the positive integers known as Fibonacci coding. Fibonacci code is a variable-length prefix code that is robust against insertion and deletion errors and is useful in data transmission and data compression. In this paper, we generalize the theorem of Zeckendorf and prove that every element of a free Z-module can be represented as a sum of elements from a Fibonacci sequence of higher order. Immediate applications of these results include a Fibonacci coding for free Z-modules, where encoding and decoding algorithms are obtained naturally from the approach of our theorems.