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 ${\displaystyle N\!}$, if ${\displaystyle d(0),d(1),\ldots ,d(k-1),d(k)\!}$ represent the digits of the code word representing ${\displaystyle N\!}$ then we have:

where F(i) is the ith Fibonacci number, and so F(i+2) is the ith distinct Fibonacci number starting with ${\displaystyle 1,2,3,5,8,13,\ldots }$. The last bit ${\displaystyle d(k)}$ 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 (that is, 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 be in, for example, 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:

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.