Let $$\mathbb {F}_q$$ F q be a finite field with q elements, $$D_{2n,\,r}$$ D 2 n , r a generalized dihedral group with $$\gcd (2n,q)=1$$ gcd ( 2 n , q ) = 1 , and $$\mathbb {F}_q[D_{2n,\,r}]$$ F q [ D 2 n , r ] a generalized dihedral group algebra. Firstly, an explicit expression for primitive idempotents of $$\mathbb {F}_q[D_{2n,\,r}]$$ F q [ D 2 n , r ] is determined, which extends the results of Brochero Martínez (Finite Fields Appl 35:204–214, 2015). Secondly, all linear complementary dual (LCD) codes and self-orthogonal codes in $$\mathbb {F}_q[D_{2n,\,r}]$$ F q [ D 2 n , r ] are precisely described and counted. Some numerical examples are also presented to illustrate our main results.

中文翻译：

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广义二面体群代数中的 LCD 码和自正交码

令 $$\mathbb {F}_q$$ F q 是一个有 q 个元素的有限域， $$D_{2n,\,r}$$ D 2 n ，ra 广义二面体群 $$\gcd (2n,q )=1$$ gcd ( 2 n , q ) = 1 和 $$\mathbb {F}_q[D_{2n,\,r}]$$ F q [ D 2 n , r ] 广义二面体群代数. 首先，确定了 $$\mathbb {F}_q[D_{2n,\,r}]$$ F q [ D 2 n , r ] 的原始幂等项的显式表达式，它扩展了 Brochero Martínez (Finite领域应用 35:204–214, 2015)。其次，$$\mathbb {F}_q[D_{2n,\,r}]$$F q [ D 2 n , r ] 中的所有线性互补对偶 (LCD) 码和自正交码都被精确描述和计数. 还提供了一些数值例子来说明我们的主要结果。