论文标题
来自组环，复合G代码和自动偶数代码结构的复合矩阵
Composite Matrices from Group Rings, Composite G-Codes and Constructions of Self-Dual Codes
论文作者
论文摘要
在这项工作中，我们定义了从组环得出的复合矩阵。我们将G代码的概念扩展到复合G代码。我们表明，这些代码是组环中的理想，其中环是有限的交换frobenius环，而G是一个任意的有限组。我们证明了复合G代码的双重二值也是复合G代码。我们定义了准复合G代码，并构建了这些代码。我们还研究了由身份矩阵和复合矩阵组成的发电机矩阵。与发电机矩阵，众所周知的扩展方法，邻居方法及其概括一起，我们发现了长度为68的极端二进制自偶代码，其中具有新的重量枚举器的稀有参数gamma = 7,8和9。特别是，我们找到了49个新代码。此外，我们表明我们发现的代码与其他构造无法访问。
In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We define quasi-composite G-codes and give a construction of these codes. We also study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters gamma=7,8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.
