On the relative profiles of a linear code and a subcode

Relative dimension/length profile (RDLP), inverse relative dimension/length profile (IRDLP) and relative length/dimension profile (RLDP) are equivalent sequences of a linear code and a subcode. The concepts were applied to protect messages from an adversary in the wiretap channel of type II with illegitimate parties. The equivocation to the adversary is described by IRDLP and upper-bounded by the generalized Singleton bound on IRDLP. Recently, RLDP was also extended in wiretap network II for secrecy control of network coding. In this paper, we introduce new relations and bounds about the sequences. They not only reveal new connections among known results but also find applications in trellis complexities of linear codes. The state complexity profile of a linear code and that of a subcode can be bounded from each other, which is particularly useful when a tradeoff among coding rate, error-correcting capability and decoding complexity is considered. Furthermore, a unified framework is proposed to derive bounds on RDLP and IRDLP from an upper bound on Communicated by T. Helleseth. Z. Zhuang · B. Dai · Y. Luo (B) Department of Computer Science and Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Min Hang District, Shanghai 200240, China e-mail: [email protected] Z. Zhuang e-mail: [email protected] Z. Zhuang · Y. Luo The State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, China B. Dai National Mobile Communications Research Laboratory, Southeast University, Nanjing, China e-mail: [email protected] A. J. Han Vinck Institute for Experimental Mathematics, Duisburg-Essen University, Essen, Germany e-mail: [email protected]