椭圆码的高效代数译码与应用

Reed-Solomon (RS)码广为应用在数字通信和存储系统，但它的码长无法超越有限域大小，纠错能力受限。本项目拟研究一套崭新的编码机制 – 椭圆码，它具备更大的码长和更强的纠错能力，获得纠错能力、纠错效率和译码复杂度之间的最佳平衡，可望在将来取代RS码。利用椭圆码进行结构性编码，我们还可获得性能优异的中短码，为下一代无线通信网络从“超高可靠”迈向“超低时延”提供技术条件。本项目将以椭圆码的设计、代数译码和应用为主线开展研究，刻画椭圆码的最小汉明距离和重量谱等代数特性，提出基于模最小化插值、渐进插值和重编码变换等前沿技术的高效代数译码，利用椭圆码进行结构性编码，为无线通信、深空通信、光纤通信和数据存储等系统提供新型编码“武器”。由于椭圆曲线广泛应用于数据加密系统，本项目的研究实现了数据加密和编码传输共享代数基础，为未来通信系统内部架构的融合创造条件。

Reed-Solomon (RS) codes have been widely used in data communication and storage systems. However, their codeword length cannot exceed the size of finite field, limiting their error-correction (EC) capability. This project investigates a new coding scheme -- elliptic codes, which have the potential of replacing RS codes in future. Comparing with RS codes defined over the same finite field, elliptic codes have larger codeword length resulting in a stronger EC capability. They can reach the optimal tradeoff between the EC capability, efficiency and complexity. Elliptic codes can also be utilized for structured coding in forging competent short-to-medium length codes. This will be the essential technique for realizing of the progression from ‘ultra-reliable’ to ‘ultra low latency’ for the next generation wireless communication networks. This project will investigate the design of elliptic codes, its algebraic decoding and applications. The code’s algebraic properties, including its minimum Hamming distance and weight spectrum, will be characterized. Its advanced algebraic decoding approaches that are empowered by module minimization interpolation, progressive interpolation and re-encoding transform will be proposed. It will also be utilized for structured coding, spinning new coding techniques for wireless communication, deep space communication, optical fiber communication and data storage systems. Since elliptic curves have been widely used for data encryption, this research enables a common algebraic foundation for both data encryption and coded transmission, facilitating the structural integration of future communication systems.