Insertion and Deletion for Involution Codes

This paper introduces a generalization of the operation of catenation: u[k]lv, the left-k-insertion, is the set of all words obtained by inserting v into u in positions that are at most k letters away from the left extremity of the word u. We define k-suffix codes using the left-kinsertion operation and extend the concept of k-prefix and k-suffix codes to involution k-prefix and involution k-suffix codes. An involution code refers to any of the generalizations of the classical notion of codes in which the identity function is replaced by an involution function. (An involution function θ is such that θ equals the identity). We also extend the notion of k-insertion closure and k-deletion closure of a language to incorporate the notion of an involution function. Thus to an involution map θ and a language L, we associate a set k-θ-ins(L) (k-θ-del(L)) with the property that their k-insertion (k-deletion) into any word of L yields words which belongs to θ(L). We study the properties of these languages.