The relative generalized Hamming weight of linear q-ary codes and their subcodes

Abstract

In this article, some properties of the relative generalized Hamming weight (RGHW) of linear codes and their subcodes are developed with techniques in finite projective geometry. The relative generalized Hamming weights of almost all 4-dimensional q-ary linear codes and their subcodes are determined.

Some bounds for the relative generalized Hamming weights of some evaluation codes

“…The study of the RGHW is motivated because of their usefulness to protect messages from an adversary in the wire-tap channel of type II with illegitimate parties. Some properties of the RGHW of q-ary codes are described in [10] and they are computed in the cases of almost all 4-dimensional linear codes and their subcodes. Moreover some equivalences, inequalities and bounds are given in [20].…”

Further results on support weights of certain subcodes

“…The value function (also called value assignment) and finite projective geometry methods are effective tools to study support weights of subcodes [2,9]. In particular, the value function was first introduced in [2] to study the GHW of a linear code, and in [7], it was generalized to study the RGHW of a linear code with respect to one of its subcodes.…”

“…In order to study the RGHW by using the value function, Liu et al [7] gave an alternative definition of the RGHW as follows.…”

A property of the relative subcodes

“…Parallel to these research work, Liu, et al [6] gave an alternative definition for the RGHW in order to develop finite projective geometry methods, that is, we can equivalently describe the RGHW as follows.…”