Research Projects

This project aims to study three problems related to Grassmann codes: (1) determining an efficient decoding algorithm, which is currently open, except for the majority voting decoder by Beelen-Singh, which does not achieve the optimum decoding capability. The project aims to continue studying Grassmann codes from a decoding perspective and find a decoder that achieves the maximum decoding radius. (2) determining generalized Hamming weights of Grassmann codes, which has been an active area of research for the last three decades. The project aims to determine new generalized Hamming weights of Grassmann codes and compute the full weight hierarchy in special cases. (3) determining the dimensions and minimum distances of linear codes associated with flag varieties over finite fields, a new and open research area. The only significant results for these codes are known from Rodier's study of line-hyperplane incidence. Although it appears challenging to determine the dimensions and minimum distances of codes associated with general flag varieties, the project is confident that it can be achieved for codes associated with two-step flag varieties.