Let $R$ be a finite commutative ring with unity $1_R$ and $k \in R$. Properties of one-sided $k$-orthogonal $n \times n$ matrices over $R$ are presented. When $k$ is idempotent, these matrices form a semigroup structure. Consequently new families of matrix semigroups over certain finite semi-local rings are constructed. When $k=1_R$, the classical orthogonal group of degree $n$ is obtained. It is proved that, if $R$ is a semi-local ring, then these semigroups are isomorphic to a finite product of $k$-orthogonal semigroups over fields. Finally, the antiorthogonal and self-orthogonal matrices that give rise to leading-systematic self-dual or weakly self-dual linear codes are discussed.
翻译:$R$ 是一个固定的流通环, 单位为 1美元, 单位为 $ 美元。 显示单方 k$- orthoonal $n\ times n$ 基质的属性, 大于 $R$ 。 当 美元为一元时, 这些基质形成一个半组结构。 因此, 在某些限定半本地环上, 新建了矩阵的组合。 当 $=1_R$ 时, 获得经典的正方位数组, 单位为 $n 。 事实证明, 如果 美元是半本地环, 这些半组是无形态的, 则在字段上, 单位为 $k$- orthogoal 半组的有限产物。 最后, 将讨论产生领先系统自制或微弱的自制线性代码的反正形和自体形矩阵矩阵。