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-1[DD)],:,{r-l,:,r-l}[TXX}][DD(X]s-l[DD)]),其中W（2r,2t,{r-1,:,r-1}[TXX}][DD(X]s-1[DD)],:,{r-l,:,r-l}[TXX}][DD(X]s-l[DD)])(∑[DD(]l[]i=1[DD)]s-i=t)表示不变因子为（2r,2t,{r-1,:,r-1}[TXX}][DD(X]s-1[DD)],:,{r-l,:,r-l}[TXX}][DD(X]s-l[DD)])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-1[DD)],:,{r-l,:,r-l}[TXX}][DD(X]s-l[DD)])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 $n(2r,2t,{r-1,:,r-1}[TXX}][DD(X]s-1[DD)],:,{r-l,:,r-l}[TXX}][DD(X]s-l[DD)])$  and the number of the orbits of $W-m(R)$, where $W(2r,2t,{r-1,:,r-1}[TXX}][DD(X]s-1[DD)],:,{r-l,:,r-l}[TXX}][DD(X]s-l[DD)])$   denotes the set of all the alternate matrices with order $m$ and the invariant factors of them are $(2r,2t,{r-1,:,r-1}[TXX}][DD(X]s-1[DD)],:,{r-l,:,r-l}[TXX}][DD(X]s-l[DD)]),$  and  $(2r,2t,{r-1,:,r-1}[TXX}][DD(X]s-1[DD)],:,{r-l,:,r-l}[TXX}][DD(X]s-l[DD)])$ denotes the number of elements in  $W(2r,2t,{r-1,:,r-1}[TXX}][DD(X]s-1[DD)],:, {r-l,:,r-l}[TXX}][DD(X]s-l[DD)]), ∑[DD(]l[]i=1[DD)]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.