Construction and enumeration of left dihedral codes satisfying certain duality properties
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Let 𝔽_q be the finite field of q elements and let D_2n=⟨ x,y| x^n=1, y^2=1, yxy=x^n-1⟩ be the dihedral group of order n. Left ideals of the group algebra 𝔽_q[D_2n] are known as left dihedral codes over 𝔽_q of length 2n, and abbreviated as left D_2n-codes. Let gcd(n,q)=1. In this paper, we give an explicit representation for the Euclidean hull of every left D_2n-code over 𝔽_q. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left D_2n-codes over 𝔽_q. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left D_2n-codes and self-dual left D_2n-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left D_2n-code over 𝔽_q, and present several numerical examples to illustrative our applications.
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