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
N
d(0),d(1),\ldots,d(k-1),d(k)
N
N=
| k-1 | |
| \sum | |
| i=0 |
d(i)F(i+2),andd(k-1)=d(k)=1.
where is the th Fibonacci number, and so is the th distinct Fibonacci number starting with
1,2,3,5,8,13,\ldots
d(k)
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.
| Symbol | Fibonacci representation | Fibonacci code word | Implied probability |
|---|---|---|---|
| 1 | F(2) | 11 | 1/4 |
| 2 | F(3) | 011 | 1/8 |
| 3 | F(4) | 0011 | 1/16 |
| 4 | F(2)+F(4) | 1011 | 1/16 |
| 5 | F(5) | 00011 | 1/32 |
| 6 | F(2)+F(5) | 10011 | 1/32 |
| 7 | F(3)+F(5) | 01011 | 1/32 |
| 8 | F(6) | 000011 | 1/64 |
| 9 | F(2)+F(6) | 100011 | 1/64 |
| 10 | F(3)+F(6) | 010011 | 1/64 |
| 11 | F(4)+F(6) | 001011 | 1/64 |
| 12 | F(2)+F(4)+F(6) | 101011 | 1/64 |
| 13 | F(7) | 0000011 | 1/128 |
| 14 | F(2)+F(7) | 1000011 | 1/128 |
To encode an integer N:
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.
Fibonacci coding has a useful property that sometimes makes it attractive in comparison to other universal codes: it is an example of a self-synchronizing code, making it easier to recover data from a damaged stream. With most other universal codes, if a single bit is altered, none of the data that comes after it will be correctly read. With Fibonacci coding, on the other hand, a changed bit may cause one token to be read as two, or cause two tokens to be read incorrectly as one, but reading a "0" from the stream will stop the errors from propagating further. Since the only stream that has no "0" in it is a stream of "11" tokens, the total edit distance between a stream damaged by a single bit error and the original stream is at most three.
This approach—encoding using sequence of symbols, in which some patterns (like "11") are forbidden, can be freely generalized.[1]
The following table shows that the number 65 is represented in Fibonacci coding as 0100100011, since . The first two Fibonacci numbers (0 and 1) are not used, and an additional 1 is always appended.
\begin{array}{ccccccccccc|c} \hline 0&1&1&2&3&5&8&13&21&34&55&-\\ \hline F(0)&F(1)&F(2)&F(3)&F(4)&F(5)&F(6)&F(7)&F(8)&F(9)&F(10)&\scriptstyleadditional\\ \hline -&-&0&1&0&0&1&0&0&0&1&1\\ \hline \end{array}
The Fibonacci encodings for the positive integers are binary strings that end with "11" and contain no other instances of "11". This can be generalized to binary strings that end with N consecutive 1's and contain no other instances of N consecutive 1's. For instance, for N = 3 the positive integers are encoded as 111, 0111, 00111, 10111, 000111, 100111, 010111, 110111, 0000111, 1000111, 0100111, …. In this case, the number of encodings as a function of string length is given by the sequence of Tribonacci numbers.
For general constraints defining which symbols are allowed after a given symbol, the maximal information rate can be obtained by first finding the optimal transition probabilities using maximal entropy random walk, then use entropy coder (with switched encoder with decoder) to encode a message as a sequence of symbols fulfilling the found optimal transition probabilities.