Vanishing Bit-Error Probability in Reed-Muller Codes

In this article, we introduce a novel approach to decoding Reed-Muller (RM) codes called camellia boosting. RM codes are widely used in digital communications due to their ability to correct errors in transmitted data. However, decoding these codes can be computationally expensive, especially when the error rate is high.
To address this challenge, we propose a new framework that leverages the concept of camellia codes. A camellia code is a type of error-correcting code that can be thought of as a collection of smaller codes, called petals, that work together to correct errors in the data. By using a camellia boosting algorithm, we can significantly reduce the computational complexity of decoding RM codes without sacrificing their error correction capabilities.
The key insight behind camellia boosting is that it only requires a weak base-case to get started, which can be obtained through a mathematical technique called entropy bending. Once the base-case is established, the algorithm can efficiently decode the RM code using a combination of linear programming and entropy decoding.
We demonstrate the effectiveness of camellia boosting by showing that it can achieve better error correction performance than existing decoding methods for RM codes. Moreover, we show that our approach can be easily extended to other types of error-correcting codes, such as q-ary symmetric input channels and q-ary RM codes.
In summary, camellia boosting is a powerful new framework for efficient decoding of RM codes that leverages the concept of camellia codes. By combining linear programming and entropy decoding, we can significantly reduce the computational complexity of decoding these codes without sacrificing their error correction capabilities. Our approach has important implications for digital communications, where accurate error correction is essential for reliable data transmission.