Symplectic code properties linked to graph matchings and hull sizes
Hull Variation in Symplectic Linear Codes
Information Theory
Summary
This paper looks at special types of error-correcting codes called symplectic linear codes. The authors introduce a way to measure an important property of these codes, called the hull dimension, using a value they call Δ(C). This measure helps them understand how the code can be changed while keeping some key characteristics the same. They also connect these code properties to concepts in graph theory, like matchings, and explain when certain optimal codes can be transformed into simpler types. This work helps clarify what hull sizes are possible in such codes.
What this means in practice
- •For coding theory engineers: Design error-correcting codes with predictable hull sizes by relating code structure to graph matching properties.
- •For quantum error correction teams: Generate symplectic codes with adjustable hull dimensions to improve code construction for quantum error correction schemes.
A theory result. No direct application yet.
Authors
Kanat Abdukhalikov, Somi Gupta, Duy Ho
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.