Symplectic Hull Control via Nonstandard Two-Coordinate Extensions and Trace-Hermitian ACD Codes

Given a linear code C over Fq with generator matrix G=[X∣Z], we study the nonstandard two-coordinate symplectic extension G¯(u,v)=[X∣Xu∣Z∣Zv] and its effects on the symplectic hull and minimum symplectic distance. The extension transforms the symplectic Gram matrix according to M¯=M+pf⊤−fp⊤, where p=Xu and f=Zv, and hence induces an alternating perturbation of rank at most two. We characterize all possible hull transitions, showing that the symplectic hull dimension can change only by −2, 0, or +2, and derive necessary and sufficient conditions for the corresponding cases. These results yield explicit criteria for symplectic LCD preservation, one-step hull reduction, a mother-code criterion for complete iterated hull reduction, and minimum-distance improvement. We further obtain exact counting formulas for admissible extension pairs, including inclusion–exclusion formulas under simultaneous structural and metric constraints, together with an automorphism-orbit reduction for eliminating equivalent extensions. In the binary case, the standard symplectic–quaternary correspondence transfers the framework to one-symbol extensions of quaternary trace-Hermitian additive complementary dual codes and yields a conditional comparison criterion with the Hermitian LCD subclass. Exhaustive computations over F2 and F3 provide finite-instance validation of the predicted hull transitions, distance behavior, and exact counts.