Hull Variation in Symplectic Linear Codes
Abstract
We study the variation of hull dimensions of symplectic linear codes under equivalence transformations preserving the symplectic distance. For a symplectic linear code C, we introduce an even-valued invariant Δ(C), which depends only on the equivalence class of C. This invariant can be expressed as the generic rank of an associated space of alternating matrices and, equivalently, as twice a corresponding matroid matching number. If C has dimension k, we prove that the minimum symplectic hull dimension among all codes equivalent to C is k−Δ(C). Consequently, C is equivalent to a symplectic LCD code if and only if Δ(C)=k. We further show that every symplectic MDS code of even dimension is monomially equivalent to a symplectic LCD code. For a symplectic self-orthogonal code C, we determine all attainable hull dimensions: for every integer r with 0≤ r≤ Δ(C)/2, there exists a monomially equivalent code whose symplectic hull has dimension k−2r. We also refine a standard form for the generator matrix of a symplectic self-orthogonal code and derive the corresponding parity-check matrix and the Gram matrix of the dual code. In the self-dual case, we characterize Δ(C) through minors of an associated symmetric matrix. Furthermore, we relate hull variation to graph matchings, establish a bound for Δ(C), and give graph constructions producing symplectic self-dual MDS, optimal, and best-known codes.
Related
- Symplectic Hull Control via Nonstandard Two-Coordinate Extensions and Trace-Hermitian ACD Codes
- Uniform Sampling of Linear Codes with Prescribed Hull Dimension
- Enumeration and Monotonicity of Linear Codes with Prescribed Hull Dimensions
- Subcodes with Prescribed Hull Dimensions in Linear Codes:Enumeration, Containment, and Asymptotic Distributions
- On the Optimal Linear Minimization Oracle Complexity of Active-Hull Methods for Minimizing Convex Hölder Smooth Functions
BibTeX
@misc{abdukhalikov2026hull,
title = {{Hull Variation in Symplectic Linear Codes}},
author = {Kanat Abdukhalikov and Somi Gupta and Duy Ho},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15535},
eprint = {2610.15535},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15535},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hull Variation in Symplectic Linear Codes
AU - Kanat Abdukhalikov
AU - Somi Gupta
AU - Duy Ho
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15535
UR - https://arxiv.org/abs/2610.15535
AB - We study the variation of hull dimensions of symplectic linear codes under equivalence transformations preserving the symplectic distance. For a symplectic linear code C, we introduce an even-valued invariant Δ(C), which depends only on the equivalence class of C. This invariant can be expressed as the generic rank of an associated space of alternating matrices and, equivalently, as twice a corresponding matroid matching number. If C has dimension k, we prove that the minimum symplectic hull dimension among all codes equivalent to C is k−Δ(C). Consequently, C is equivalent to a symplectic LCD code if and only if Δ(C)=k. We further show that every symplectic MDS code of even dimension is monomially equivalent to a symplectic LCD code. For a symplectic self-orthogonal code C, we determine all attainable hull dimensions: for every integer r with 0≤ r≤ Δ(C)/2, there exists a monomially equivalent code whose symplectic hull has dimension k−2r. We also refine a standard form for the generator matrix of a symplectic self-orthogonal code and derive the corresponding parity-check matrix and the Gram matrix of the dual code. In the self-dual case, we characterize Δ(C) through minors of an associated symmetric matrix. Furthermore, we relate hull variation to graph matchings, establish a bound for Δ(C), and give graph constructions producing symplectic self-dual MDS, optimal, and best-known codes.
ER -