A new kind of geometric structures determining the chain good weight hierarchies (Q1861247)

The paper gives some new sufficient conditions to determine the weight hierarchies of linear codes satisfying the chain condition. The concept of weight hierarchy of a linear code was introduced by \textit{T. Helleseth, T. Kløve}, and \textit{J. Mykkeltveit} [Discrete Math. 18, 179-211 (1977; Zbl 0357.94007)]. Later on \textit{V. K. Wei} [IEEE Trans. Inf. Theory 37, 1412-1418 (1991; Zbl 0735.94008)] showed the relationship between the weight hierarchy and the problem of the wiretap channel of type II and \textit{T. Kasami, T. Takata, T. Fujiwara}, and \textit{S. Lin} [IEEE Trans. Inf. Theory 39, 242-245 (1993; Zbl 0765.94014)] showed the importance of the weight hierarchy in the analysis of the trellis complexity of linear codes. The chain condition was introduced by \textit{V. K. Wei} and \textit{K. Yang} [IEEE Trans. Inf. Theory 39, 1709-1713 (1993; Zbl 0801.94016)]. In the present paper the weight hierarchy \((d_1,d_2,\dots,d_k)\) of a code \({\mathcal C}\) is called chain good if \({\mathcal C}\) satisfies the chain condition. Following previous ideas of \textit{W. Chen} and \textit{T. Kløve} [Designs Codes Cryptography 15, 47-66 (1998; Zbl 0913.94010)] the authors give a geometric structure that allows them to prove some new sufficient conditions so that a given sequence \((a_1,a_2,\dots,a_k)\) of positive integers is a chain good weight hierarchy.