Optimal Code Pairs Satisfying Relative Chain Condition

The relative generalized Hamming weight (RGHW) of a linear code and a linear subcode is a generalization of generalized Hamming weight (GHW). It characterizes the code performance of the wiretap channel of type II, secure network coding, linear ramp secret sharing scheme and trellis complexity. In this paper, we present a new proof of the Griesmer bound on RGHW by the idea of residue code. Motivated by the proof, we introduce the concept of relative chain condition and show that code pairs meeting the Singleton, weak Plotkin and Griesmer bounds satisfy this condition. This research may be helpful for studying the connection between RGHW and trellis complexity, the weight hierarchy of code pairs, etc.