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

Yuan CAO  Yonglin CAO  Jian GAO  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E100-A   No.12   pp.2585-2593
Publication Date: 2017/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E100.A.2585
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Coding Theory and Techniques
left dihedral code,  finite chain ring,  self-dual code,  LCD code,  

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Let $mathbb{F}_q$ be a finite field of q elements, $R=mathbb{F}_q+umathbb{F}_q$ (u2=0) and D2n=<x, y | xn=1, y2=1, yxy=x-1> be a dihedral group of order n. Left ideals of the group ring R[D2n] are called left dihedral codes over R of length 2n, and abbreviated as left D2n-codes over R. Let n be a positive factor of qe+1 for some positive integer e. In this paper, any left D2n-code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes Ai and outer codes Ci, where Ai is a cyclic code over R of length n and Ci is a linear code of length 2 over a Galois extension ring of R. More precisely, a generator matrix for each outer code Ci is given. Moreover, a formula to count the number of these codes is obtained, the dual code for each left D2n-code is determined and all self-dual left D2n-codes over R are presented, respectively.