Symplectic Hull Control via Nonstandard Two-Coordinate Extensions and Trace-Hermitian ACD Codes
- 1College of Computer Engineering, Jimei University, Xiamen 361021, China
- 2School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin Institute of Technology, Harbin 150001, China
Abstract
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.
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BibTeX
@article{cheng2026symplectic,
title = {{Symplectic Hull Control via Nonstandard Two-Coordinate Extensions and Trace-Hermitian ACD Codes}},
author = {Kangda Cheng and Weilin Wu and Xiaobao Li and Jinlong Liu},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14214063},
url = {https://www.mdpi.com/2227-7390/14/21/4063},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Symplectic Hull Control via Nonstandard Two-Coordinate Extensions and Trace-Hermitian ACD Codes
AU - Kangda Cheng
AU - Weilin Wu
AU - Xiaobao Li
AU - Jinlong Liu
PY - 2026
DA - 2026/11/05
JO - Mathematics
DO - 10.3390/math14214063
UR - https://www.mdpi.com/2227-7390/14/21/4063
AB - 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.
SN - 2227-7390
ER -