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Negafibonacci coding

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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.

Encoding method

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The following steps describe how to encode a nonzero integer . Note that denotes the Negafibonacci sequence.

  1. If is positive, compute the greatest odd negative integer such that the sum of the odd negative terms of the Negafibonacci sequence from -1 to with a step of -2, is greater than or equal to :

    If is negative, compute the greatest even negative integer such that the sum of the even negative terms of the Negafibonacci sequence from 0 to with a step of -2, is less than or equal to :
  2. Add a 1 at the bit of the binary word. Subtract from .
  3. Repeat the process from step 1 with the new value of x, until it reaches 0.
  4. Add a 1 on the left of the resulting binary word to finish the encoding.

To decode an encoded binary word, remove the leftmost 1 from the binary word, since it is used only to denote the end of the encoded number. Then assign the remaining bits the values of the Negafibonacci sequence from -1 (1, −1, 2, −3, 5, −8, 13...), and sum the all the values associated with a 1.

Negafibonacci representation

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Negafibonacci coding 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.

Table

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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

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References

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Works cited

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  • Knuth, Donald (2008). Negafibonacci Numbers and the Hyperbolic Plane. 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. Addison-Wesley. ISBN 978-0-321-58050-4. 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.