A finite group with a cyclic normal subgroup N such that G/N is cyclic is said to be metacyclic. A code over a finite field F is a metacyclic code if it is a left ideal in the group algebra FG for G a metacyclic group. Metacyclic codes are generalizations of dihedral codes, and can be constructed as quasi-cyclic codes with an extra automorphism. In this paper, we prove that metacyclic codes form an asymptotically good family of codes. Our proof relies on a version of Artin's conjecture for primitive roots in arithmetic progression being true under the Generalized Riemann Hypothesis (GRH).

中文翻译：

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元循环码的渐近性能

具有循环正规子群 N 使得 G/N 是循环的有限群称为元循环。如果有限域 F 上的代码是元循环群 G 的群代数 FG 中的左理想，则该代码是元循环代码。元循环码是二面体码的推广，可以构造为具有额外自同构的准循环码。在本文中，我们证明了元循环码形成了一个渐近良好的码族。我们的证明依赖于阿廷猜想的一个版本，即在广义黎曼假设 (GRH) 下，算术级数中的原始根为真。