OpenAlex ·
Hull Variation in Symplectic Linear Codes
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2610.15535
- Original
- Read it at the publisher
- Topics
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 $0\le r\le Δ(C)/2$, 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.
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