环Z/p\+kZ上m阶交错矩阵的计数定理及其应用

设W\-m(R)是有限局部环R=Z/p\+kZ上所有m阶交错矩阵所构成的集合（p是素数,k>1）. 该文通过确定R上任意m阶交错矩阵的标准形,计算出W\-m(R)在线性群GL\-m(R)作用下的轨道数及n(2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:,\{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)]),其中W（2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:,\{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)])(∑DD(]l]i=1DD)]s\-i=t)表示不变因子为（2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:,\{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)])的所有m阶交错矩阵构成的集合,n(2r,2t,（2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:,\{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)])表示其中的元素个数. 最后,作者利用有限局部环R上交错矩阵的标准形构作了一个Cartesian认证码,并计算出其全部参数.

Some Anzahl Theorems of Alternate Matrices over Z/pkZ and its Application

Let \$W\-m(R)\$  be the set of alternate matrices over \$Z/p\+kZ\$ with order  \$m\$, where \$m≥2,p\$ is aprime and \$k>1\$. By determining the normal form of alternate matrices over\$Z/p\+kZ,\$ the compute \$n(2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:,\{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)])\$  and the number of the orbits of \$W\-m(R)\$, where \$W(2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:,\{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)])\$   denotes the set of all the alternate matrices with order \$m\$ and the invariant factors of them are \$(2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:,\{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)]),\$  and  \$(2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:,\{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)])\$ denotes the number of elements in  \$W(2r,2t,\{r\-1,\:,r\-1\}TXX}]DD(X]s\-1DD)],\:, \{r\-l,\:,r\-l\}TXX}]DD(X]s\-lDD)]), ∑DD(]l]i=1DD)]s\-i=t. \$ Furthermore, using the normal form of alternate matrices, the authors construct a Cartesian authentication code and compute the parameters of Cartesian authentication code.