On the RDLP and IRDLP of a linear code and a linear subcode

For a linear code and a linear subcode, the relative dimension/length profile (RDLP), inverse relative dimension/ length profile (IRDLP) and relative length/dimension profile (RLDP) are three character sequences, which have been applied to the wiretap channel of type II with illegitimate parties and extended in the wiretap network II for the secrecy control of linear network coding. They also provide useful information to analyze the trellis complexity of a linear code. These concepts are two-code generalizations of the DLP, IDLP and LDP proposed by Forney, respectively. The Singleton bounds on RDLP, IRDLP and RLDP also extend Forney´s bounds. In this paper, we introduce new relations with respect to RDLP and IRDLP. We show that the Singleton bounds on RDLP, IRDLP and RLDP can be derived directly from these relations. Some interesting duality properties of RDLP and IRDLP are also given.