Adaptive Base Representation Theorem: An Alternative to Binary Number System
Ravin Kumar
TL;DR
The paper tackles the problem of representing integers with $n$ bits in a storage-density parity equivalent to binary, by introducing the Adaptive Base Representation (ABR). ABR defines a per-position base sequence $\{B_i\}$ and a signed representation $v = \sum_{i=0}^{n-1} (-1)^{\epsilon(i)} d_i \times B_i$, with $d_i \in \{0,1\}$ and an $\epsilon(i)$ rule that prevents over-counting. The central theoretical result proves the equivalence $\mathbb{B}_n = \mathbb{A}_n$ for all $n$, ensuring a unique ABR representation for every $v$ and compatibility with standard coding and error-correction tools such as Huffman coding, arithmetic coding, and Hamming codes. The work demonstrates ABR's potential for digital steganography and broad implications for information theory and mathematics, with concrete base-value analyses and example encodings.
Abstract
This paper introduces the Adaptive Base Representation (ABR) Theorem and proposes a novel number system that offers a structured alternative to the binary number system for digital computers. The ABR number system enables each decimal number to be represented uniquely and using the same number of bits, $n$, as the binary encoding. Theoretical foundations and mathematical formulations demonstrate that ABR can encode the same integer range as binary, validating its potential as a viable alternative. Additionally, the ABR number system is compatible with existing data compression algorithms like Huffman coding and arithmetic coding, as well as error detection and correction mechanisms such as Hamming codes. We further explore practical applications, including digital steganography, to illustrate the utility of ABR in information theory and digital encoding, suggesting that the ABR number system could inspire new approaches in digital data representation and computational design.