Bounds on relative generalised Hamming weight

Abstract

The rth relative generalised Hamming weight (RGHW) of an [ n , k ] linear code C and an [ n , k 1 ] subcode C 1 , a generalisation of generalised Hamming weight (GHW), characterises code performances of wiretap channel of type II, secure network coding, linear ramp secret sharing scheme, trellis complexity etc. In this study, the authors investigate non-asymptotic and asymptotic bounds on RGHW. In the non-asymptotic setting, they present a new proof of the Griesmer bound on RGHW by residue code and introduce the new concept of relative chain condition. They show that code pairs meeting the Griesmer, Singleton, and weak Plotkin bounds satisfy this condition. The notion of relative chain condition and these results provide a new perspective on researches including evaluating trellis complexity and determining the weight hierarchy of code pairs etc. In the asymptotic setting, they improve previous work by introducing two new metrics, respectively, for the cases r is fixed and r is proportionally increasing with n. They show the asymptotic Singleton, Plotkin, and Gilbert–Varshamov bounds on the first metric and determine the value of the second metric, which is helpful for characterising the optimal asymptotic performances of applications and constructing the optimal coding schemes.