Code constructions and existence bounds for relative generalized Hamming weight

The relative generalized Hamming weight (RGHW) of a linear code C and a subcode C ¹ is an extension of generalized Hamming weight. The concept was firstly used to protect messages from an adversary in the wiretap channel of type II with illegitimate parties. It was also applied to the wiretap network II for secrecy control of network coding and to trellis-based decoding algorithms for complexity estimation. For RGHW, bounds and code constructions are two related issues. Upper bounds on RGHW show the possible optimality for the applications, and code constructions meeting upper bounds are for designing optimal schemes. In this article, we show indirect and direct code constructions for known upper bounds on RGHW. When upper bounds are not tight or constructions are hard to find, we provide two asymptotically equivalent existence bounds about good code pairs for designing suboptimal schemes. Particularly, most code pairs (C, C ¹) are good when the length n of C is sufficiently large, the dimension k of C is proportional to n and other parameters are fixed. Moreover, the first existence bound yields an implicit lower bound on RGHW, and the asymptotic form of this existence bound generalizes the usual asymptotic Gilbert–Varshamov bound.