Negafibonacci coding

In mathematics, negaFibonacci coding is a universal code which encodes nonzero integers into binary code words. It is similar to Fibonacci coding, except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end. The code for the integers from -11 to 11 is given below.

xx  negaFibonacci representation            negaFibonacci code
-11  101000                                 0001011
-10  101001                                 1001011
-9   100010                                 0100011
-8   100000                                 0000011
-7   100001                                 1000011
-6   100100                                 0010011
-5   100101                                 1010011
-4   1010                                   01011
-3   1000                                   00011
-2   1001                                   10011
-1   10                                     011
0    0                                      (cannot be encoded)
1    1                                      11
2    100                                    0011
3    101                                    1011
4    10010                                  010011
5    10000                                  000011
6    10001                                  100011
7    10100                                  001011
8    10101                                  101011
9    1001010                                01010011
10   1001000                                00010011
11   1001001                                10010011

The Fibonacci code is closely related to negaFibonacci representation, a positional numeral system sometimes used by mathematicians. The negaFibonacci code for a particular nonzero integer is exactly that of the integer's negaFibonacci representation, except with the order of its digits reversed and an additional "1" appended to the end. The negaFibonacci code for all negative numbers has an odd number of digits, while those of all positive numbers have an even number of digits.

To encode a nonzero integer X:

1. Calculate the largest (or smallest) encodeable number with N bits by summing the odd (or even) negafibonacci numbers from 1 to N.
2. When it is determined that N bits is just enough to contain X, subtract the Nth negaFibonacci number from X, keeping track of the remainder, and put a one in the Nth bit of the output.
3. Working downward from the Nth bit to the first one, compare each of the corresponding negaFibonacci numbers to the remainder. Subtract it from the remainder if the absolute value of the difference is less, AND if the next higher bit does not already have a one in it. A one is placed in the appropriate bit if the subtraction is made, or a zero if not.
4. Put a one in the N+1th bit to finish.

To decode a token in the code, remove the last "1", assign the remaining bits the values 1,-1,2,-3,5,-8,13... (the negafibonacci numbers), and add the "1" bits.

See also

• Golden ratio base
• Zeckendorf's theorem

References

• Knuth, Donald (2008), Negafibonacci Numbers and the Hyperbolic Plane, Paper presented at the annual meeting of the Mathematical Association of America, San Jose, California.
• Knuth, Donald (2009), The Art of Computer Programming, Volume 4, Fascicle 1: Bitwise Tricks & Techniques; Binary Decision Diagrams, ISBN 0-321-58050-8. In the pre-publication draft of section 7.1.3 see in particular pp. 36–39.
• Margenstern, Maurice (2008), Cellular Automata in Hyperbolic Spaces, Advances in unconventional computing and cellular automata, vol. 2, Archives contemporaines, p. 79, ISBN 9782914610834.