New asymptotic metrics for relative generalized Hamming weight

It was recently shown that RGHW (relative generalized Hamming weight) exactly expresses the security of linear ramp secret sharing scheme. In this paper we determine the true value of the asymptotic metric for RGHW previously proposed by Zhuang et al. in 2013. Then we propose new asymptotic metrics useful for investigating the optimal performance of linear ramp secret sharing scheme constructed from a pair of linear codes. We also determine the true values of the proposed metrics in many cases.

I. Introduction

It was shown in [2], [4] that any pair of nested linear codes can be used for constructing a linear ramp secret sharing scheme [1], [9]. Recently, Kurihara et al. [6] showed that the smallest number of shares required for an adversary to illegitimately obtain at least bits of information is exactly expressed by the -th relative generalized Hamming weight (RGHW) of proposed by Luo et al. [7]. In order to clarify how good secret sharing schemes can be constructed from linear codes, it is indispensable to discover existential bounds on RGHW similar to the Gilbert-Varshamov bound. Toward this direction, Zhuang et al. [10] and this author [8] proposed such existential bounds.