群环中零因子码和单位码
Codes from Zero-divisors and Units in Group Rings
摘要：
我们描述并提出了一种新的构造方法的编码使用群环。它们主要包括两种类型：零因子和单位导出码。以前的群环码集中在理想上;例如循环码是循环群上的群环中的理想。新的焦点是编码本身，只有在非常有限的条件下才能产生理想。我们利用群环同构于某个定义良好的矩阵环的结果，从而每个群环元素都有一个相关矩阵。这使得矩阵代数可以根据需要用于代码的研究和生成，从而能够创建标准生成器和校验矩阵。群环是单位和零因子的丰富来源，由此产生新的代码。许多代码属性，例如LDPC或自对偶，可以表示为群环内的属性，从而使得能够构造具有这些属性的代码。这些方法是通用的，使得能够构造具有许多类型的群环的码。对环没有限制，因此整数上的码，矩阵环甚至群环本身都是可能的和富有成效的。
英文摘要：
We describe and present a new construction method for codes using encodings from group rings. They consist primarily of two types: zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; for example cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in ideals. We use the result that a group ring is isomorphic to a certain well-defined ring of matrices, and thus every group ring element has an associated matrix. This allows matrix algebra to be used as needed in the study and production of codes, enabling the creation of standard generator and check matrices. Group rings are a fruitful source of units and zero-divisors from which new codes result. Many code properties, such as being LDPC or self-dual, may be expressed as properties within the group ring thus enabling the construction of codes with these properties. The methods are general enabling the construction of codes with many types of group rings. There is no restriction on the ring and thus codes over the integers, over matrix rings or even over group rings themselves are possible and fruitful.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
