Type-II polyadic constacyclic codes over finite fields

Abstract

Polyadic constacyclic codes over finite fields have been of interest due to their nice algebraic structures, good parameters, and wide applications. Recently, the study of Type-I [Formula: see text]-adic constacyclic codes over finite fields has been established. In this paper, a family of Type-II [Formula: see text]-adic constacyclic codes is investigated. The existence of such codes is determined using the length of orbits in a suitable group action. A necessary condition and a sufficient condition for a positive integer [Formula: see text] to be a multiplier of a Type-II [Formula: see text]-adic constacyclic code are determined. Subsequently, for a given positive integer [Formula: see text], a necessary condition and a sufficient condition for the existence of Type-II [Formula: see text]-adic constacyclic codes are derived. In many cases, these conditions become both necessary and sufficient. For the other cases, determining necessary and sufficient conditions is equivalent to the discrete logarithm problem which is considered to be computationally intractable. Some special cases are investigated together with examples of Type-II polyadic constacyclic codes with good parameters.