  In this work, we propose two criteria for linear codes obtained from the
Plotkin sum construction being symplectic self-orthogonal (SO) and linear
complementary dual (LCD). As specific constructions, several classes of
symplectic SO codes with good parameters including symplectic maximum distance
separable codes are derived via $\ell$-intersection pairs of linear codes and
generalized Reed-Muller codes. Also symplectic LCD codes are constructed from
general linear codes. Furthermore, we obtain some binary symplectic LCD codes,
which are equivalent to quaternary trace Hermitian additive complementary dual
codes that outperform best-known quaternary Hermitian LCD codes reported in the
literature. In addition, we prove that symplectic SO and LCD codes obtained in
these ways are asymptotically good.
