Search

Let F 2 m be a finite field of cardinality 2 m , R = F 2 m [ u ] ∕ 〈 u 4 〉 = F 2 m + u F 2 m + u 2 F 2 m + u 3 F 2 m (u 4 = 0) which is a finite chain ring, and n is an odd positive integer. For any δ , α ∈ F 2 m × , an explicit representation for the dual code of any (δ + α u 2) -constacyclic code over R of length 2 n is given. And some dual codes of (1 + u 2) -constacyclic codes over R of length 14 are constructed. For the case of δ = 1 , all distinct self-dual (1 + α u 2) -constacyclic codes over R of length 2 n are determined. [ABSTRACT FROM AUTHOR]

Let D 2 n = 〈 x , y ∣ x n = 1 , y 2 = 1 , y x y = x − 1 〉 be a dihedral group, and R = GR ( p 2 , m ) be a Galois ring of characteristic p 2 and cardinality p 2 m where p is a prime. Left ideals of the group ring R [ D 2 n ] are called left dihedral codes over R of length 2 n , and abbreviated as left D 2 n -codes over R . Let gcd ( n , p ) = 1 in this paper. Then any left D 2 n -code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A i and outer codes C i , where A i is a cyclic code over R of length n and C i is a skew cyclic code of length 2 over a Galois ring or principal ideal ring extension of R . Specifically, a generator matrix and basic parameters for each outer code C i are given. A formula to count the number of these codes is obtained and the dual code for each left D 2 n -code is determined. Moreover, all self-dual left D 2 n -codes and self-orthogonal left D 2 n -codes over R are presented. [ABSTRACT FROM AUTHOR]

Abstract Let F 2 m be a finite field of cardinality 2 m , n be an odd positive integer, and denote R = F 2 m [ u ] ∕ 〈 u 3 〉. Let δ , α ∈ F 2 m ×. Then (δ + α u 2) -constacyclic codes over R are called constacyclic codes over R of Type 2. In this paper, an explicit representation and a complete description for all distinct (δ + α u 2) -constacyclic codes over R of length 2 n 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 (1 + α u 2) -constacyclic codes over R of length 2 n are presented precisely. In addition, a complement to a result in Cao et al. (2017) is given. [ABSTRACT FROM AUTHOR]