有限半局部环上的单边$k$ -正交矩阵及其码
One-Sided $k$-Orthogonal Matrices Over Finite Semi-Local Rings And Their Codes
摘要：
设$R$是具有单位元$1_R$和$k \in R$的有限交换环.给出了$R$上单侧$k$ -正交$n \times n$矩阵的性质.当$k$是幂等的时，这些矩阵形成半群结构。从而构造了有限半局部环上的矩阵半群的新族。当$k=1_R$时，得到了$n$次的经典正交群.证明了如果$R$是半局部环，则这些半群同构于域上的$k$ -正交半群的有限积.最后，讨论了导致系统自对偶或弱自对偶线性码的反正交矩阵和自正交矩阵。
英文摘要：
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.
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