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Zeckendorf's Theorem and Fibonacci Coding for Modules. (arXiv:1706.06655v1 [cs.IT])
来源于:arXiv
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 $\mathbb{Z}$-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 $\mathbb{Z}$-modules, where
encoding and decoding algorithms are obtained naturally from the approach of
our theorems. 查看全文>>