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Concept
Fibonacci coding
Summary
In mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no other instances of "11" before the end.
The Fibonacci code is closely related to the Zeckendorf representation, a positional numeral system that uses Zeckendorf's theorem and has the property that no number has a representation with consecutive 1s. The Fibonacci code word for a particular integer is exactly the integer's Zeckendorf representation with the order of its digits reversed and an additional "1" appended to the end.
For a number , if represent the digits of the code word representing then we have:
where F(i) is the ith Fibonacci number, and so F(i+2) is the ith distinct Fibonacci number starting with . The last bit is always an appended bit of 1 and does not carry place value.
It can be shown that such a coding is unique, and the only occurrence of "11" in any code word is at the end i.e. d(k−1) and d(k). The penultimate bit is the most significant bit and the first bit is the least significant bit. Also leading zeros cannot be omitted as they can in e.g. decimal numbers.
The first few Fibonacci codes are shown below, and also their so-called implied probability, the value for each number that has a minimum-size code in Fibonacci coding.
To encode an integer N:
Find the largest Fibonacci number equal to or less than N; subtract this number from N, keeping track of the remainder.
If the number subtracted was the ith Fibonacci number F(i), put a 1 in place i−2 in the code word (counting the left most digit as place 0).
Repeat the previous steps, substituting the remainder for N, until a remainder of 0 is reached.
Place an additional 1 after the rightmost digit in the code word.
To decode a code word, remove the final "1", assign the remaining the values 1,2,3,5,8,13... (the Fibonacci numbers) to the bits in the code word, and sum the values of the "1" bits.
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