The r th relative generalised Hamming weight (RGHW) of an linear code C and an subcode , 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.

中文翻译：

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相对广义汉明权重的界

的 [R 的相对广义汉明权重（RGHW） 线性码 C 和 子码 ，是广义汉明权重（GHW）的概括，它描述了II型窃听通道的代码性能，安全网络编码，线性斜坡秘密共享方案，网格复杂性等。 。在非渐近设置中，他们通过残基代码提供了Griesmer结合到RGHW上的新证明，并引入了相对链条件的新概念。他们表明满足Griesmer，Singleton和弱Plotkin边界的代码对满足此条件。相对链条条件的概念和这些结果为研究提供了新的视角，包括评估网格复杂性和确定代码对的权重等级等。在渐近设置中，他们分别通过引入两个新指标来改进以前的工作，[R 是固定的 [R 与成比例增加 ñ 。它们在第一个度量上显示渐近的Singleton，Plotkin和Gilbert-Varshamov边界，并确定第二个度量的值，这有助于表征应用程序的最佳渐近性能并构造最佳编码方案。