On (t, L)-fold perfect authentication and secrecy codes with arbitration

An authentication code with arbitration is t-fold perfect if the numbers of decoding rules and encoding rules meet the information-theoretic lower bounds with equality. A code has perfect L-fold secrecy if observing a set of $$L'\le L$$L′≤L messages in the channel gives no information to the opponent regarding the $$L'$$L′ source states. In this paper, we investigate (t, L)-fold perfect authentication and secrecy codes with arbitration which provide both t-fold perfect and perfect L-fold secrecy. We define a new design, L-secrecy perfect ordered restricted strong partially balanced t-design, which is used to construct a (t, L)-fold perfect authentication and secrecy code with arbitration. We also obtain some infinite classes of (t, 1)-fold perfect authentication and secrecy codes with arbitration, especially for $$t>2$$t>2.