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