We develop a system theory for left dihedral codes only using finite field theory, basic theory of cyclic codes and skew cyclic codes.
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We prove that any left dihedral code is a direct sum of concatenated codes in which the inner codes and outer codes are cyclic codes and skew cyclic codes respectively.
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We provide a precise expression for all distinct left -code over , where is the dihedral group of order n and .
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We give the dual code of any left -code and list all self-dual left -codes explicitly.
Abstract
Let be the dihedral group of order n. Left ideals of the group algebra are known as left dihedral codes over of length 2n, and abbreviated as left -codes. In this paper, a system theory for left -codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left -code is a direct sum of concatenated codes with inner codes and outer codes , where is a minimal self-reciprocal cyclic code over of length n and is a skew cyclic code of length 2 over an extension field or principal ideal ring of . Then for the case of , we give a precise description for outer codes in the concatenated codes, provide the dual code for any left -code and determine all self-dual left -codes. Moreover, all 1995 binary left dihedral codes and all 255 binary self-dual left dihedral codes of length 30 are given, and a class of left -codes over is investigated.