This paper focuses on hull dimensional codes obtained by the intersection of linear codes and their dual. These codes were introduced by Assmus and Key and have been the subject of significant theoretical and practical research over the years, gaining increased attention in recent years. Let \(\mathbb {F}_q\) denote the finite field with q elements, and let G be a finite Abelian group of order n. The paper investigates Abelian codes defined as ideals of the group algebra \(\mathbb {F}_qG\) with coefficients in \(\mathbb {F}_q\). Specifically, it delves into Abelian hull dimensional codes in the group algebra \(\mathbb {F}_qG\), where G is a finite Abelian group of order n with \(\gcd (n,q)=1\). Specifically, we first examine general hull Abelian codes and then narrow its focus to Abelian one-dimensional hull codes. Next, we focus on Abelian one-dimensional hull codes and present some necessary and sufficient conditions for characterizing them. Consequently, we generalize a recent result on Abelian codes and show that no binary or ternary Abelian codes with one-dimensional hulls exist. Furthermore, we construct Abelian codes with one-dimensional hulls by generating idempotents, derive optimal ones with one-dimensional hulls, and establish several existing results of Abelian codes with one-dimensional hulls. Finally, we develop enumeration results through a simple formula that counts Abelian codes with one-dimensional hulls in \(\mathbb {F}_qG\). These achievements exploit the rich algebraic structure of those Abelian codes and enhance and increase our knowledge of them by considering their hull dimensions, reducing the gap between their interests and our understanding of them.

本文重点介绍了线性码与它们的对偶交集得到的船体维度码。这些代码由 Assmus 和 Key 引入，多年来一直是重要理论和实践研究的主题，近年来受到越来越多的关注。设 \（\mathbb {F}_q\） 表示具有 q 个元素的有限域，设 G 是 n 阶的有限阿贝尔群。本文研究了定义为群代数 \（\mathbb {F}_qG\） 的理想状态的阿贝尔码，系数为 \（\mathbb {F}_q\）。具体来说，它深入研究了代数群 \（\mathbb {F}_qG\） 中的阿贝尔船体维度码，其中 G 是 n 阶的有限阿贝尔群，其中 \（\gcd （n，q）=1\）。具体来说，我们首先研究一般的船体阿贝尔代码，然后将其关注范围缩小到阿贝尔一维船体代码。接下来，我们关注阿贝尔一维船体代码，并提出一些必要和充分的条件来表征它们。因此，我们推广了最近关于 Abelian 码的结果，并表明不存在具有一维外壳的二进制或三元 Abelian 码。此外，我们通过生成幂等函数来构建具有一维外壳的阿贝尔码，用一维外壳推导出最优码，并建立了具有一维外壳的阿贝尔码的几个现有结果。最后，我们通过一个简单的公式来计算枚举结果，该公式计算 \（\mathbb {F}_qG\） 中具有一维外壳的阿贝尔码。 这些成就利用了这些阿贝尔代码丰富的代数结构，并通过考虑它们的外壳尺寸来增强和增加我们对它们的了解，缩小它们的兴趣和我们对它们的理解之间的差距。