Transversals, Orthogonality, and Enumeration in Linear Cyclic Latin Squares

We study the two-parameter class of linear cyclic arrays L(x,y) = ax + by (mod N) and describe its basic combinatorial structure in a unified arithmetic way. We determine the exact maximum size of a pairwise orthogonal subfamily within this class, showing that it is exactly p_min(N) - 1, where p_min(N) is the smallest prime divisor of N. While cyclic and group-based orthogonality phenomena are classical, our contribution is a short arithmetic treatment that also includes explicit criteria for Latinity, a parity-based proof of transversal existence, a classification and exact enumeration of affine transversals, explicit odd-order transversal decompositions, and an exact multiplicative formula for the number of linear orthogonal mates. In particular, within this class the even-order obstruction is absolute on the transversal side: in even order there are no transversals at all. On the orthogonality side, the maximum pairwise orthogonal subfamily size collapses to 1.