We define additive G-codes over finite fields. We prove that if C is an additive G-code over ${\mathbb{F}}_q$ with duality M then its dual with respect to this duality $C^M$ is an additive G-code. We prove that if M and $M^{\prime }$ are two dualities, then $C^M$ and $C^{M^{\prime }}$ are equivalent codes. Finally, we study the existence of self-dual codes for a variety of dualities and relate them to formally self-dual and linear self-dual codes.

中文翻译：

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加性G代码${\mathbb{F}}_q$ 及其二重性

我们在有限域上定义加性G码。我们证明如果C是加性G代码${\mathbb{F}}_q$具有对偶M，则关于该对偶的对偶$C^{\mathrm{中号}}$是加性G代码。我们证明如果M和${\mathrm{中号}}^{\prime }$ 是两个对偶 $C^{\mathrm{中号}}$ 和 $C^{{\mathrm{中号}}^{\prime }}$是等效代码。最后，我们研究了多种对偶性的自对偶码的存在，并将它们与形式上的自对偶和线性自对偶码相关联。