A Class of Left Dihedral Codes Over Rings $\mathbb{F}_q+u\mathbb{F}_q$

Abstract

Let $\mathbb{F}_q$ be a finite field of q elements, $R=\mathbb{F}_q+u\mathbb{F}_q$ (u²=0) and D_2n=<x, y | xⁿ=1, y²=1, yxy=x^-1> be a dihedral group of order n. Left ideals of the group ring R[D_2n] are called left dihedral codes over R of length 2n, and abbreviated as left D_2n-codes over R. Let n be a positive factor of q^e+1 for some positive integer e. In this paper, any left D_2n-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 linear code of length 2 over a Galois extension ring of R. More precisely, a generator matrix for each outer code C_i is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D_2n-code is determined and all self-dual left D_2n-codes over R are presented, respectively.