Symplectic Hull-Variation Analysis

• The symplectic hull-variation problem is a framework that studies how the dimension and structure of symplectic hulls change under coordinate permutations and group actions.
• It employs both algebraic and enumerative methods, using symplectic isometries and mass formulas to analyze invariant properties and orbit structures in codes over finite fields and non-unital rings.
• The analysis supports optimal code construction and classification while addressing open challenges like joint enumeration by hull-dimension and minimum distance.

The symplectic hull-variation problem is a central topic in the theory of linear codes endowed with symplectic forms, addressing how the dimension and structure of symplectic hulls respond to coordinate permutations and group actions. Originally studied over finite fields and, more recently, over exotic algebraic structures such as non-unital rings, this problem captures intricate relationships between code invariants, group symmetries, and mass enumeration. Analysis of the symplectic hull-variation problem provides both structural and enumerative insight into symplectic codes, with consequences for classification, equivalence, and applications such as optimal code construction.

1. Algebraic Framework for Symplectic Codes

Symplectic hull-variation arises in the context of linear codes over module or vector spaces equipped with a symplectic form. For codes over the non-unital ring

$$E = \langle \kappa, \tau \mid 2\kappa=2\tau=0,\, \kappa^2=\kappa,\, \tau^2=\tau,\, \kappa\tau=\kappa,\, \tau\kappa=\tau \rangle,$$

$E$ has four elements $\{0,\, \kappa,\, \tau,\, \zeta=\kappa+\tau\}$, is characteristic 2, and does not possess a multiplicative identity. The unique maximal (nilpotent) ideal is $J = \{0, \zeta\}$, so $E/J \cong \mathbb{F}_2$. Codes are left $E$-submodules $C \subset E^{2n}$, with the free case defined by coincidence of the residue and torsion codes, i.e., $C_{(\mathrm{Res})}=C_{(\mathrm{Tor})}$, where the residue and torsion codes are binary subcodes extracted via projection modulo $\zeta$ or by requiring divisibility by $\zeta$.

For codes over a field $\mathbb{F}_q$, the ambient space is $V=\mathbb{F}_q^{2n}$. Symplectic forms are given by the standard block matrix $$J_{2n} = \begin{pmatrix} 0 & I_n \ -I_n & 0 \end{pmatrix}$$, and the symplectic inner product by $$(x, y)_s = x J_{2n} y^T = \sum_{i=1}^n (x_i y_{n+i} - x_{n+i} y_i)$$.

2. Symplectic Hulls and Their Duals

Given a code $C$, the left, right, and two-sided symplectic duals are defined by:

• $$C^{\perp_{S_L}} = \{\,z \mid \langle z, w \rangle_s = 0\ \forall\,w\in C\,\}$$,
• $$C^{\perp_{S_R}} = \{\,z \mid \langle w, z \rangle_s = 0\ \forall\,w\in C\,\}$$,
• $C^{\perp_s} = C^{\perp_{S_L}} \cap C^{\perp_{S_R}}$.

Their associated hulls are

• $\mathrm{LSHull}(C) = C \cap C^{\perp_{S_L}}$,
• $\mathrm{RSHull}(C) = C \cap C^{\perp_{S_R}}$,
• $\mathrm{SHull}(C) = C \cap C^{\perp_s}$.

In the field case, the symplectic hull is $\Hull_s(C) = C \cap C^{\perp_s}$, with $h_s(C) = \dim_{\mathbb{F}_q}\Hull_s(C)$ always an even integer satisfying $0 \le h_s(C) \le k$ and $k-h_s(C)$ even. Codes with $\mathrm{SHull}(C) = \{0\}$ are called symplectic LCD (linear complementary dual) codes.

3. Formulation of the Symplectic Hull-Variation Problem

The symplectic hull-variation problem investigates the variation in hull rank under permutations of code coordinates. For a free $E$-linear $[2n, k]$ code $C$, and for a permutation matrix $P$ of size $2n \times 2n$, the object of study is the set

$$\mathcal{V}(C) = \{\mathrm{rank}_E(\mathrm{SHull}(C P)) : P \text{ a permutation matrix}\}.$$

The principal questions are:

• For which $P$ does $$\mathrm{rank}\,\mathrm{SHull}(C P) = \mathrm{rank}\,\mathrm{SHull}(C)$$ hold?
• What is the range of possible ranks of the symplectic hull as $P$ ranges over all coordinate permutations?
• How do invariants like $n$, $k$, or the initial hull rank control these variations?

This formulation generalizes naturally to codes over finite fields, where analogous questions are posed with respect to field-automorphism-induced permutations or the action of the full symplectic group.

4. Symmetry, Isometries, and Invariance Criteria

A central result is that the symplectic hull rank is invariant under permutations $P$ precisely when $P$ preserves the symplectic form, i.e.,

$P \Omega_{2n} P^T = \Omega_{2n}.$

Here, $\Omega_{2n}$ is the symplectic block matrix determining the inner product. Such $P$ form the symplectic group $Sp(2n, \mathbb{F}_2)$ or, correspondingly, in field generality, $Sp(2n, q)$. For $Q \in Sp(2n, q)$, the code $CQ$ satisfies

$(CQ)^{\perp_{s}} = C^{\perp_{s}} Q, \quad \Hull_s(CQ) = \Hull_s(C) Q,$

and thus hull-dimension is preserved.

In contrast, for coordinate permutations $P$ with $P \Omega P^T \neq \Omega$, genuine hull-variation occurs: the rank $\mathrm{rank}(\mathrm{SHull}(CP))$ can increase or decrease, as demonstrated by explicit examples over $E$ and $\mathbb{F}_2$ of length 4, where the hull-rank drops from 2 to 0 after a swap operation not preserving $\Omega$. Permutation-equivalence of codes, in the absence of symplectic isometry, is therefore insufficient for hull-invariance.

A concise summary of invariance properties:

Condition on $P$	Hull Rank Invariance	Group Type
$P\Omega P^T = \Omega$	Yes	Symplectic isometries
General $P$	Not in general	Arbitrary permutations

5. Enumeration, Mass Formulas, and Orbit Structure

For codes over finite fields, the set of all $[2n, k]$ codes with prescribed hull-dimension $r$ forms an orbit under the symplectic group $Sp(2n, q)$. The result of Li–Shi–Li–Ling provides a closed mass formula for the cardinality:

$$M_q(n, k, r) = q^{2k_0(n-k_0)} \binom{n}{k_0}_{q^2} \prod_{i=1}^r \frac{q^{2n-(2k_0 + r)-(i-1)} - q^{r-1}}{(q^r - 1)q^{2k_0 + (i-1)}}$$

where $k = 2k_0 + r$ and $\binom{n}{k_0}_{q^2}$ denotes the Gaussian binomial coefficient. Codes with $r=0$ (symplectic LCD) constitute a unique symplectic group orbit; general $r$-hull codes partition accordingly by the invariant $h_s(C) = r$.

This enumerative approach is underpinned by the orbit-stabilizer theorem and is constructive: starting from a canonical LCD code, one accounts for extensions by symplectic self-orthogonal vectors to reach higher hull-dimensions, with recurrence relations for the counts.

Concrete evaluations for small parameters, such as $q=2$, $n=2$, $k=2$, yield explicit numbers: $M_2(2, 2, 0) = 20$, $M_2(2, 2, 2) = 15$. The overall code count $\binom{4}{2}_2 = 35$ matches the sum. Similar explicit data is accessible for other regimes.

6. Asymptotic Analysis and Dominance of Symplectic LCD Codes

As $n \to \infty$ with fixed $k_0$, $r$, asymptotic behaviors emerge:

$$\lim_{n \to \infty} \frac{M_q(n, k, r)}{q^{2k_0(n-k_0)} \binom{n}{k_0}_{q^2}} = \frac{1}{(q^r-1)^r\, g_{q^2, \infty}}$$

where $$g_{q^2, \infty} = \prod_{i=1}^\infty (1 - q^{-2i}) > 0$$. The dominant contribution is the “LCD part,” while the hull-dimension $r$ appears only in the multiplicative prefactor. This suggests that, for large length, the vast majority of codes (on the order of $q^{2k_0(n-k_0)}$) are symplectic LCD, and codes with large hulls are exceedingly rare.

7. Applications, Open Problems, and Research Directions

Analysis of the symplectic hull-variation problem yields practical consequences for the classification of codes over $E$ and over finite fields. In particular, the ability to enumerate and characterize codes with specified hull-dimension underpins the classification of optimal free $E$-linear codes for small lengths and provides benchmarks for code equivalence under both the permutation group and the larger symplectic group.

Open avenues of research identified include:

• Mass enumeration for codes up to monomial equivalence, in contrast to permutation or symplectic group actions.
• Extension of mass formulas to Hermitian or Euclidean hulls, especially over extension fields such as $\mathbb{F}_{q^2}$.
• Joint enumeration by both hull-dimension and minimum distance, which remains largely unexplored.

The findings of these works provide the symplectic analog in a broader hull-variation framework, setting a foundation for further exploration of invariants and equivalence classes in symplectic and related code families (Kushwaha et al., 10 Jan 2026, Li et al., 2024).