Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes
Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes
The relative generalized Hamming weight of linear q-ary codes and their subcodes
Designs, Codes and Cryptography
The bit extraction problem or t-resilient functions
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Geometric approach to higher weights
IEEE Transactions on Information Theory - Part 1
The worst case probability of undetected error for linear codes on the local binomial channel
IEEE Transactions on Information Theory
Efficient maximum likelihood decoding of linear block codes using a trellis
IEEE Transactions on Information Theory
Some new characters on the wire-tap channel of type II
IEEE Transactions on Information Theory
Generalized Hamming weights for linear codes
IEEE Transactions on Information Theory
Generalized Hamming weights of linear codes
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Codes satisfying the chain condition
IEEE Transactions on Information Theory
Dimension/length profiles and trellis complexity of linear block codes
IEEE Transactions on Information Theory
Upper bounds on generalized distances
IEEE Transactions on Information Theory
Bounds on the minimum support weights
IEEE Transactions on Information Theory
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The relative generalized Hamming weight (RGHW) of a linear code C and a subcode C1 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, C1) 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.