Type 2 constacyclic codes over [Formula presented] of oddly even length

Abstract

© 2018 Elsevier B.V. Let [Formula presented] be a finite field of cardinality [Formula presented], [Formula presented] be an odd positive integer, and denote [Formula presented]. Let [Formula presented]. Then [Formula presented]-constacyclic codes over [Formula presented] are called constacyclic codes over [Formula presented] of Type 2. In this paper, an explicit representation and a complete description for all distinct [Formula presented]-constacyclic codes over [Formula presented] of length [Formula presented] and their dual codes are given. Moreover, explicit formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct self-dual [Formula presented]-constacyclic codes over [Formula presented] of length [Formula presented] are presented precisely. In addition, a complement to a result in Cao et al. (2017) is given.