// Fibonacci Coding - Self-synchronizing code using golden ratio mathematics
Works for any positive integer without parameters.
Can recover from transmission errors using '11' pattern.
Based on Fibonacci sequence and Zeckendorf theorem.
Fibonacci coding uses Zeckendorf's theorem: every positive integer has a unique representation as a sum of non-consecutive Fibonacci numbers. The code consists of this representation in binary (1 if Fibonacci number is used, 0 if not) followed by an additional 1 as terminator. The '11' pattern only appears at code end.
Fibonacci: 1, 2, 3, 5, 8, 13, 21... 1 = F(1) → 11 2 = F(2) → 011 3 = F(3) → 0011 4 = F(3)+F(1) → 1011 5 = F(4) → 00011 12 = F(5)+F(3)+F(1) → 101011 No consecutive Fibonacci numbers used '11' only appears at code end
Fibonacci coding is a universal code that represents positive integers using the Fibonacci sequence. It's based on Zeckendorf's theorem and creates self-synchronizing codes where '11' only appears as a terminator.
Zeckendorf's theorem states that every positive integer can be uniquely represented as a sum of non-consecutive Fibonacci numbers. This representation forms the basis of Fibonacci coding.
The pattern '11' (two consecutive ones) only appears at the end of each codeword. This allows a decoder to resynchronize after transmission errors by looking for '11' patterns.
Fibonacci coding is used in data compression research, error-resilient transmission systems, and theoretical computer science. It's valued more for its mathematical properties than practical efficiency.