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 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.

Encoding method

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.

Table

The code for the integers from −11 to 11 is given below.

number	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

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, 2, Archives contemporaines, p. 79, ISBN 9782914610834.