Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
In data compression, a universal code for integers is a prefix code that maps the positive integers onto binary codewords, with the additional property that whatever the true probability distribution on integers, as long as the distribution is monotonic , the expected lengths of the codewords are...
which encodes positive integers into binary code word
Code word
In communication, a code word is an element of a standardized code or protocol. Each code word is assembled in accordance with the specific rules of the code and assigned a unique meaning...
s. Each code word ends with "11" and contains no other instances of "11" before the end.
Definition
For a number , if represent the digits of the code word representing then we have:
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
.
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 code begins as follows:
Symbol
Fibonacci representation
Fibonacci code word
1
11
2
011
3
0011
4
1011
5
00011
6
10011
7
01011
8
000011
The Fibonacci code is closely related to the Zeckendorf representation, a positional numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
Zeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers....
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.
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
equal to or less than N; subtract this number from N, keeping track of the remainder.
If the number subtracted was the ith Fibonacci number F(i), put a 1 in place i−2 in the code word (counting the left most digit as place 0).
Repeat the previous steps, substituting the remainder for N, until a remainder of 0 is reached.
Place an additional 1 after the rightmost digit in the code word.
To decode a code word, remove the final "1", assign the remaining the values 1,2,3,5,8,13... (the Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
s) to the bits in the code word, and sum the values of the "1" bits.
Comparison with other universal codes
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
Self-synchronizing code
In telecommunications, a self-synchronizing code is a line code in which the symbol stream formed by a portion of one code word, or by the overlapped portion of any two adjacent code words, is not a valid code word...
, making it easier to recover data from a damaged stream. With most other universal codes, if a single bit
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
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
Edit distance
In information theory and computer science, the edit distance between two strings of characters generally refers to the Levenshtein distance. However, according to Nico Jacobs, “The term ‘edit distance’ is sometimes used to refer to the distance in which insertions and deletions have equal cost and...
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 http://aps.arxiv.org/pdf/0710.3861.
Example
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.
0
1
1
2
3
5
8
13
21
34
55
–
additional
–
–
0
1
0
0
1
0
0
0
1
1
A method to encode any integer is shown in the following Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
program.
def encode_fib(n):
# Return string with Fibonacci encoding for n (n >= 1).
result = ""
if n >= 1:
a = 1
b = 1
c = a + b # next Fibonacci number
fibs = [b] # list of Fibonacci numbers, starting with F(2), each <= n
while n >= c:
fibs.append(c) # add next Fibonacci number to end of list
a = b
b = c
c = a + b
result = "1" # extra "1" at end
for fibnum in reversed(fibs):
if n >= fibnum:
n = n - fibnum
result = "1" + result
else:
result = "0" + result
return result
Golden ratio base is a non-integer positional numeral system that uses the golden ratio as its base. It is sometimes referred to as base-φ, golden mean base, phi-base, or, colloquially, phinary...
Zeckendorf's theorem, named after Belgian mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers....
In data compression, a universal code for integers is a prefix code that maps the positive integers onto binary codewords, with the additional property that whatever the true probability distribution on integers, as long as the distribution is monotonic , the expected lengths of the codewords are...
The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.
, if 
represent the digits of the code word representing 
then we have:
