Abstract:

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.

Key words: Normal form of alternate matrix, Anzahl theorems, Orbit, Finite local ring, Authentication code