Speaker
Description
Determining the minimum Hamming distance of an error-correcting code $\mathcal{C}$ has long stood as a fundamental challenge in coding theory. In this talk, we turn our attention to an even more ambitious problem: determining the full Hamming weight distribution of $\mathcal{C}$. This means counting the number of codewords in $\mathcal{C}$ of each possible Hamming weight—an essential but notoriously difficult task, especially for general codes.
While computing the weight distribution for an arbitrary code remains largely out of reach, the problem becomes more tractable under additional structural assumptions, especially when the code $\mathcal{C}$ is cyclic. We will discuss the calculation of three families of cyclic codes with arbitrarily many nonzeroes, highlighting its connections to linear algebra, graph theory, and association schemes.
The talk is based on joint work with Gennian Ge (Capital Normal University), Sihuang Hu (Shandong University), Tao Feng (Zhejiang University); as well as Maosheng Xiong (Hong Kong University of Science and Technology) and Haode Yan (Harbin Institute of Technology, Shenzhen).