arXiv:2605.14823v4 Announce Type: replace Abstract: In this paper, a class of linear authentication codes with secrecy, which are equipped with simple encoding rules, is constructed. By means of a special Weil sum, the maximum success probabilities of impersonation attack (denoted by $P_I$) and of substitution attack (denoted by $P_S$) for these codes are explicitly derived. It is further proven that the codes are asymptotically optimal, in the sense that both the information theoretic bound and the combinatorial bound for authentication codes are asymptotically attained by $P_I$ and $P_S$.