Construction and enumeration of left dihedral codes satisfying certain duality properties |
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Affiliation: | 1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China;2. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China |
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Abstract: | Let be the finite field of q elements and let be the dihedral group of 2n elements. Left ideals of the group algebra are known as left dihedral codes over of length 2n, and abbreviated as left -codes. Let . In this paper, we give an explicit representation for the Euclidean hull of every left -code over . On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left -codes over . In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left -codes and self-dual left -codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left -code over , and present several numerical examples to illustrative our applications. |
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Keywords: | Left dihedral code Euclidean duality Hull of a linear code Self-dual code LCD code Self-orthogonal code |
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