Hermitian ℓ-Complementary Codes

• Hermitian ℓ-complementary codes are linear error-correcting codes over 𝔽₍q²₎ defined by a Hermitian hull of dimension ℓ, unifying self-orthogonal, LCD, and self-dual codes.
• They are constructed using algebraic methods, matrix scaling, and algebraic geometry, yielding explicit families with optimal or near-optimal parameters.
• These codes are essential in developing entanglement-assisted quantum error-correcting codes and advancing classical coding theory, with active research on higher-dimensional extensions.

A Hermitian $\ell$-complementary code is a linear code $C \leq \mathbb{F}_{q^2}^n$ whose Hermitian hull—defined via the non-degenerate Hermitian inner product—is of prescribed dimension $\ell$ ($0 \leq \ell \leq k$). These codes substantively generalize Hermitian self-orthogonal, Hermitian linear complementary dual (LCD), and Hermitian self-dual codes, and their structural understanding is central to advances in both classical and quantum coding theory.

1. Definitions and Fundamental Properties

Let $\mathbb{F}_{q^2}$ be a finite field with involutive automorphism $x \mapsto x^q$, and let $C \leq \mathbb{F}_{q^2}^n$ be a linear $[n, k]_{q^2}$ code. The Hermitian inner product is

$\langle x, y \rangle_H = \sum_{i=1}^n x_i y_i^q,$

for $x, y \in \mathbb{F}_{q^2}^n$. The Hermitian dual is $$C^{\perp_H} := \{ v \in \mathbb{F}_{q^2}^n \mid \langle v, c\rangle_H = 0\ \forall c\in C \}$$, and the Hermitian hull is $Hull_H(C) := C \cap C^{\perp_H}$, with dimension $\ell = \dim_{\mathbb{F}_{q^2}} Hull_H(C)$. A code is Hermitian $\ell$-complementary if $\dim Hull_H(C) = \ell$.

Special cases include:

• $\ell = 0$: Hermitian LCD code ($C \cap C^{\perp_H} = 0$)
• $\ell = k$: Hermitian self-orthogonal code ($C \subseteq C^{\perp_H}$)
• $\ell = k = n/2$: Hermitian self-dual code ($C = C^{\perp_H}$)

For matrix-based constructions, a $k \times n$ generator matrix $G$ satisfies $\dim Hull_H(C) = \dim(\ker(G\bar{G}^T))$, where $\bar{G}$ is entrywise conjugation.

2. Algebraic and Combinatorial Constructions

Explicit constructions of Hermitian $\ell$-complementary codes can be founded on the principle that for any Hermitian self-orthogonal $[n, k, d]_{q^2}$ code, there exists for every $0 \leq \ell \leq k$ an $[n, k, d]_{q^2}$ code $C^{(\ell)}$ with $\dim Hull_H(C^{(\ell)}) = \ell$. This is achieved by scaling columns of a systematic generator matrix $G = [I_k \,|\, A]$ by non-square elements of $\mathbb{F}_{q^2}^*$, with rank arguments ensuring the desired hull dimension. Such techniques enable the construction of codes with arbitrary hull dimension starting from Hermitian self-orthogonal seeds (Sok, 2021).

Algebraic geometry (AG) evaluation codes from curves of genus $g$ further provide families of Hermitian self-orthogonal codes: the AG code $C_L(D, G; v)$ constructed from a divisor $G$ of large degree on a curve $X/\mathbb{F}_{q^2}$, under explicit conditions on $G$ and the differential $\omega$, is Hermitian self-orthogonal with dimension and hull determined by the curve and divisor properties. Particular instances include MDS codes from the projective line ($g=0$), almost-MDS codes from elliptic curves ($g=1$), and very long codes from Hermitian curves ($g=q(q-1)/2$) (Sok, 2021).

For codes over $\mathbb{F}_4$, column-by-column classification yields a canonical generator matrix for Hermitian LCD codes of dimension $2$, enabling exhaustive search and classification (Ishizuka, 2020).

3. Enumeration and Asymptotic Properties

The enumeration of Hermitian $\ell$-complementary codes is captured by $N_{n,k}^{(\ell)}(q)$: the number of $[n, k]_{q^2}$ codes with Hermitian hull of dimension $\ell$. For $\ell = k$ (self-orthogonal), an explicit product formula depending on $n$'s parity is known. For general $\ell$, inclusion-exclusion via Möbius inversion in the lattice of subspaces yields: $$N_{n, k}^{(\ell)}(q) = \sum_{s = \ell}^{\lfloor n/2 \rfloor} \sigma(n,s) \binom{s}{\ell}_{q^2} \binom{n-2s}{k-s}_{q^2} (-1)^{s-\ell} q^{(s-\ell)(s-\ell-1)},$$ where $\sigma(n, s)$ is the number of $[n, s]$ Hermitian self-orthogonal codes, and the Gaussian binomials are over $\mathbb{F}_{q^2}$ (Wang et al., 14 Dec 2025).

Asymptotically for large $q$, the average weight enumeration of Hermitian self-orthogonal codes matches that of all unrestricted codes. Moreover, for fixed $d$, the fraction of Hermitian self-orthogonal codes of dimension $k$ with minimum distance less than $d$ vanishes as $q \to \infty$, provided $k \leq n-d+1$. It follows that Hermitian self-orthogonal MDS codes are asymptotically dense in the space of all such codes as $q$ grows (Wang et al., 14 Dec 2025).

4. Structural and Codeword Characterizations

A codeword-level criterion for the Hermitian LCD property asserts that for a linear code $C$ over $\mathbb{F}_{q^2}$, $C$ is Hermitian LCD if and only if for every nonzero $c_1 \in C$ there exists $c_2 \in C$ with $\langle c_1, c_2 \rangle_H \neq 0$. This is equivalent to the invertibility of $G\bar{G}^T$ for a generator matrix $G$. For one-generator $\ell$-quasi-cyclic codes, necessary and sufficient algebraic conditions reduce to (i) an invariance under Frobenius and reciprocal map for the polynomial generator $g(x)$ and (ii) coprimality of a certain folded sum of the generating polynomials with the co-generator (Guan et al., 2023).

For small dimensions, canonical generator matrices and combinatorial conditions (e.g., parity constraints on the multiplicities of column types) completely determine existence and optimality, as in the classification of optimal quaternary Hermitian LCD codes of dimension $2$ (Ishizuka, 2020) and the nonexistence theorems for $k \geq 3$ (Araya et al., 2019).

5. Explicit Families and Parameter Ranges

Construction methods yield explicit infinite families of Hermitian $\ell$-complementary codes with optimal or near-optimal parameters, including:

• MDS and almost-MDS codes from AG curves and systematic matrix generators with hull dimension varying over the full allowed range (Sok, 2021);
• Quaternary Hermitian LCD codes with new record minimum distances via extension-and-puncturing of generator matrices (Harada, 2021);
• Complete classification of optimal $[n,2,d]_4$ Hermitian LCD codes for all $n$ except those with $n \equiv 4 \pmod{5}$, and explicit families for $k=3$ (Ishizuka, 2020, Araya et al., 2019).

Parameter sets for constructed codes include families of MDS $[q^2+1, k, n-k+1]_{q^2}$ codes for projective lines, almost-MDS codes from elliptic/hyperelliptic curves, and high-dimensional codes $k > (n+q-1)/(q+1)$ which were previously inaccessible for prescribed hull dimension (Sok, 2021).

6. Applications to Quantum Error Correction

Hermitian $\ell$-complementary codes are instrumental in constructing entanglement-assisted quantum error correcting codes (EAQECCs). Given a $[n, k, d]_{q^2}$ code with hull dimension $\ell$, the standard construction yields

$$[[n,\, k-\ell,\, d;\, n-k-\ell]]_q \quad \text{and} \quad [[n,\, n-k-\ell,\, d^\perp;\, k-\ell]]_q,$$

where $d^\perp$ is the minimum distance of the Hermitian dual (Sok, 2021, Araya et al., 2019). Codes constructed via the algebraic geometric and systematic generator methods can attain the EA-Singleton bound for a wide range of parameters. For example, MDS Hermitian LCD codes can be used to construct maximal-entanglement EAQECCs meeting the Griesmer bound when $k=3$, providing distance-optimal quantum codes (Araya et al., 2019).

7. Open Problems and Future Directions

Challenges remain in extending classification and construction to higher dimensions, systematically closing gaps in the table of optimal parameters for Hermitian LCD and more general $\ell$-complementary codes. The generalization of combinatorial bounds on column multiplicities and the discovery of new highly-structured seed codes for propagation via extension remain open. On the quantum side, it is unresolved whether maximal-entanglement EAQECCs meeting the Griesmer bound can always be achieved for arbitrary $k \geq 4$ (Araya et al., 2019, Harada, 2021).

Ongoing research addresses both the explicit identification of new families with prescribed hull and the asymptotic landscape, where almost all self-orthogonal codes are MDS as $q$ grows. Classification for $k > 3$ and extension of algebraic curve constructions to new parameter ranges are active areas of investigation (Wang et al., 14 Dec 2025, Sok, 2021).