Abstract: We got the standard form of nonsingular matrix over the finite field by means of BN pair decomposition under the similarity transf ormation of the permutation matrix. A construction method of Cartesian authentic ation codes is presented and the parameters are computed. Moreover, assuming that the encoding rules are chosen, P_IandP_S, which denote the largest uniformly distributed probability of a successful impersonation attack of these codes and that of a successful substitution attack respectively, are also computed.

Key words: authentication codes, general linear group, BN pair decomposition