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Asymptotic performance of metacyclic codes
Discrete Mathematics ( IF 0.7 ) Pub Date : 2020-07-01 , DOI: 10.1016/j.disc.2020.111885
Martino Borello , Pieter Moree , Patrick Solé

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:

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。
更新日期:2020-07-01
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