A few days ago, Associate Professor Liu Zihui from the School of Mathematics and Statistics of Beijing Institute of Technology and his graduate student Wei Yao made new results in the study of the relative generalized Hamming weight of codes, and published them online in the authoritative journal of information theory IEEE Transactions on Information Theory.

The relatively generalized Hamming weight is an important parameter given by Luo and Vinck et al. in 2005 to describe the code for secure information transmission. This parameter describes the minimum cost that a channel eavesdropper must pay to eavesdrop on the remaining information when part of the information in the channel is leaked. After this concept was put forward, it has been one of the hotspots of theoretical research in information theory and coding.

Previous studies have given a relatively generalized theoretical upper bound of Hamming weight. The significance of this theoretical upper bound is that the coding scheme that reaches the upper bound can make the information transmitted safely as much as possible. The work done by predecessors has given the proof of the theoretical upper bound, but the shortcoming of the work is that the theoretical coding scheme that reaches the upper bound has not been constructed.

Associate Professor Liu Zihui and his collaborators successfully introduced the projective geometry method on the finite field to study the relatively generalized Hamming weight, and gave a new method to prove the upper bound of the relatively generalized Hamming weight theory through the counting technique of the subspace in the projective space. Furthermore, through the in-depth promotion of this brand-new proof method on the finite field, a coding scheme construction method of any dimension that reaches the upper bound of the relatively generalized Hamming weight theory is also given. The construction method of the coding scheme not only enables the maximum safe transmission of information, but at the same time, its significance lies in determining the weight distribution of the code for the low-dimensional code, especially the minimum distance of the code, that is, the error correction ability of the code.

The coding scheme constructed by the new achievements has the largest information security transmission, is convenient to analyze the decoding error probability and has a strong theoretical error correction ability and many theoretical advantages.

Link to the paper: DOI:10.1109/TIT.2021.3078064

About the author:

Liu Zihui is an associate professor in the School of Mathematics and Statistics, Beijing Institute of Technology. He has long been engaged in the research of coding theory and information security. He has published more than 40 papers on IEEE Transactions on Information Theory, IEEE Communications Letters, Finite Fields and Their applications, Designs Codes and Cryptography, SIAM Journal on Discrete Mathematics, Science China, Discrete Mathematics, and other authoritative journals, and has published 2 papers in the well-known international conference ISIT in the field of information theory.