Asymptotic Weight Distribution

• Asymptotic Weight Distribution is an analysis of the statistical behavior of Hermitian ℓ-complementary codes, defined using Gaussian binomial coefficients and weight enumerators.
• The methodology incorporates exact counting formulas and asymptotic approximations that highlight the prevalence of MDS self-orthogonal codes in the large field limit.
• Algebraic constructions, including scaling, puncturing, and algebraic geometry techniques, provide practical frameworks for designing both classical codes and entanglement-assisted quantum error-correcting codes.

A Hermitian $\ell$-complementary code is a linear code over $\mathbb{F}_{q^2}^n$ whose intersection with its Hermitian dual (the Hermitian hull) has prescribed dimension $\ell$, where $0 \leq \ell \leq k$ for an $[n,k]_{q^2}$ code. This concept subsumes Hermitian self-orthogonal codes ($\ell = k$), Hermitian LCD (linear complementary dual) codes ($\ell = 0$), and intermediate hull dimensions. The theory of Hermitian $\ell$-complementary codes is grounded in the structure of the unitary space over finite fields, with deep applications in both classical and quantum coding theory, especially for constructing optimal classical codes and entanglement-assisted quantum error-correcting codes (EAQECCs).

1. Formal Definitions and Notation

Let $C \leq \mathbb{F}_{q^2}^n$ be a $k$-dimensional linear code. The Hermitian inner product on $\mathbb{F}_{q^2}^n$ is defined as

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

where $x, y \in \mathbb{F}_{q^2}^n$. The Hermitian dual of $C$ is

$$C^{\perp_H} = \{ v \in \mathbb{F}_{q^2}^n : \langle v, c \rangle_H = 0 \ \forall c \in C \}.$$

The Hermitian hull is

$\mathrm{Hull}_H(C) = C \cap C^{\perp_H},$

and the hull dimension is $\ell = \dim_{\mathbb{F}_{q^2}} \mathrm{Hull}_H(C)$, satisfying $0 \leq \ell \leq k$. $C$ is called:

• Hermitian LCD if $\ell=0$;
• Hermitian self-orthogonal if $\ell = k$ and $C \subseteq C^{\perp_H}$;
• Hermitian self-dual if $\ell = k = n/2$.

These codes are collectively termed Hermitian $\ell$-complementary codes. This nomenclature is standard in recent literature (Wang et al., 14 Dec 2025, Sok, 2021).

2. Enumerative and Asymptotic Properties

The enumeration and statistical distribution of Hermitian $\ell$-complementary codes are central to understanding their prevalence and optimality, especially in the large field limit.

Exact Counting

The number of $[n,k]_{q^2}$ codes with Hermitian hull dimension $\ell$ is denoted $N_{n,k}^{(\ell)}(q)$. Let $w = \lfloor n/2 \rfloor$. When $\ell = k$ (self-orthogonal case), one has: $$N_{n,k}^{(k)}(q) = \begin{cases} \prod_{i=1}^k \frac{(q^{n-2i+2}-1)(q^{n-2i+1}+1)}{q^{2i}-1} & \text{if } n \text{ even,}\ \prod_{i=1}^k \frac{(q^{n-2i+2}+1)(q^{n-2i+1}-1)}{q^{2i}-1} & \text{if } n \text{ odd.} \end{cases}$$ For general hull dimension $\ell$ with $0\leq \ell \leq k$,

$$N_{n,k}^{(\ell)}(q) = \sum_{s=\ell}^{\lfloor n/2 \rfloor} \sigma(n,s) \binom{s}{\ell}_{q^2} \binom{n-2s}{k-s}_{q^2} (-1)^{s-\ell} q^{(s-\ell)(s-\ell-1)},$$

where $\sigma(n,s)$ is as above, and $\binom{a}{b}_{q^2}$ is the Gaussian binomial coefficient (Wang et al., 14 Dec 2025).

Asymptotic Behavior

For fixed $n, k$, the average weight distribution $\bar A_i(n,k,k)$ of Hermitian self-orthogonal $[n,k]$ codes approaches that of unrestricted $[n,k]$ codes as $q\to\infty$: $\bar A_i(n,k,k) \sim \binom{n}{i} q^{2(i-n+k)}.$ Most Hermitian self-orthogonal codes are MDS ($d=n-k+1$) in the large-$q$ limit, and for codes in $\Sigma(n,k,k)$ (the set of all such codes), the probability that $d(C)=n-k+1$ tends to $1$ as $q\to\infty$ (Wang et al., 14 Dec 2025).

3. Algebraic and Explicit Constructions

Hermitian $\ell$-complementary codes can be constructed or manipulated via several explicit algebraic techniques:

From Hermitian Self-Orthogonal to Arbitrary Hull Dimension

Given any Hermitian self-orthogonal $[n,k,d]_{q^2}$ code $C$, for every $0 \leq \ell \leq k$ there exists a code $C^{(\ell)}$ with $\dim \mathrm{Hull}_H(C^{(\ell)}) = \ell$. This is achieved by scaling $k-\ell$ columns of a systematic-form generator matrix by a nonsquare $\gamma \in \mathbb{F}_{q^2}^*$ (Sok, 2021).

Algebraic Geometry Curve Constructions

Hermitian self-orthogonal codes are built from algebraic-geometric codes over curves $X/\mathbb{F}_{q^2}$ of genus $g$ with $n$ rational points. By picking divisors $G$ and differential forms $\omega$ with suitable conditions (e.g., $k > (n+q+2g-1)/(q+1)$), one constructs codes with desirable hull dimensions and parameters. Explicit families include:

• Projective line (g=0): Gives MDS codes $[q^2+1,k,q^2-k+2]_{q^2}$.
• Elliptic curves (g=1): Yield almost-MDS codes.
• Hermitian curves (g=$q(q-1)/2$): Yield long MDS codes.

These produce codes with hull dimension $0$ (Hermitian LCD), but hull-reduction techniques yield the whole range of $\ell$ (Sok, 2021).

Puncturing and Embedding Methods

Puncturing certain zero columns of an MDS Hermitian self-orthogonal code's generator matrix modifies the hull dimension, allowing for prescribed $\ell$. In the projective line and higher-genus cases, this framework produces MDS or almost-MDS codes with a full spectrum of hull dimensions (Sok, 2021).

4. Characterization and Structure of Hermitian LCD and $\ell$-Complementary Codes

A code $C\leq \mathbb{F}_{q^2}^n$ is Hermitian LCD (linear complementary dual), i.e., $\ell = 0$, if $C \cap C^{\perp_H} = \{0\}$. A generator matrix $G$ realizes this property if and only if $G \bar{G}^T$ is nonsingular, with $\bar{G}$ the field involution applied entrywise (Harada, 2021, Ishizuka, 2020).

For quasi-cyclic codes, $C$ is Hermitian LCD precisely when:

1. The generator polynomial $g(x)$ satisfies $g(x) = \tilde{g}^q(x)$ (where $\tilde{g}(x)$ is the reciprocal polynomial),
2. The folded sum $\sum_{i=0}^{\ell-1} f_i(x) \bar{f}_i^q(x)$ is coprime to $(x^n-1)/g(x)$ (Guan et al., 2023).

The codeword-level LCD property is characterized by: For every nonzero $c_1 \in C$, there exists $c_2 \in C$ such that $\langle c_1, c_2 \rangle_H \neq 0$ (Guan et al., 2023).

5. Classification, Existence, and Optimality in Special Cases

Low-Dimensional Quaternary Hermitian LCD Codes

For $k=2$ over $\mathbb{F}_4$, every $[n,2]$ Hermitian LCD code can be put in a canonical generator-matrix form, classified up to equivalence by the vector multiplicities of five canonical column types. The maximal minimum distance is

$$d_4(n,2) = \begin{cases} \lfloor 4n/5 \rfloor, & n \equiv 1,2,3 \pmod{5}, \ \lfloor 4n/5 \rfloor - 1, & n \equiv 0,4 \pmod{5}. \end{cases}$$

For $k=3$, $d_4(n,3)$ and explicit constructions are similarly given, with the table structure of generator matrices and block-type enumeration playing a pivotal role (Ishizuka, 2020, Araya et al., 2019).

Extension and Puncturing Techniques

Hermitian LCD codes can be extended by adding two coordinates while preserving the LCD property. Puncturing certain coordinates or shortening retains the LCD property provided nonsingularity of $G\bar G^T$ is maintained (Harada, 2021).

Nonexistence Results

For certain parameters $(n,k,d)$, Hermitian LCD codes cannot exist, e.g., no $[4^{k-1} s, k, 4^{k-1} s]$ codes exist for $k \geq 3, s \geq 1$, due to combinatorial block-multiplicity bounds (Araya et al., 2019).

6. Applications to Quantum Error Correction

Hermitian $\ell$-complementary codes, particularly LCD and self-orthogonal codes, are instrumental in constructing entanglement-assisted quantum error-correcting codes (EAQECCs). The hull dimension directly controls the amount of required entanglement. Given an $[n,k,d]_{q^2}$ code with Hermitian hull dimension $\ell$, there exist two EAQECCs:

• $$\left[\!\left[n,\,k-\ell,\,d;\,n-k-\ell\right]\!\right]_q$$,
• $$\left[\!\left[n,\,n-k-\ell,\,d^\perp;\,k-\ell\right]\!\right]_q$$,

where $d^\perp$ is the minimum distance of the dual code. If the input code is MDS and $d \leq n+1$, then the EAQECC achieves the EA-Singleton bound $2(d-1) = n+c-k$ (Sok, 2021).

Explicit families of Hermitian LCD and self-orthogonal codes thus yield optimal quantum codes, including maximal-entanglement EAQECCs matching the entanglement-assisted Griesmer bound in tabulated infinite families (Araya et al., 2019).

7. Ongoing Directions and Open Problems

Key ongoing questions and themes include:

• Systematic enumeration and construction of Hermitian $\ell$-complementary codes for larger fields and dimensions, especially optimizing the minimum distance for given length and dimension (Wang et al., 14 Dec 2025).
• Strengthening nonexistence bounds for Hermitian LCD codes using combinatorial design theory and improving block-multiplicity arguments for higher $k$ (Araya et al., 2019).
• Exploring extensions and puncturing strategies to generate longer code families and to close existing gaps in the minimum distance tables for small lengths (Harada, 2021).
• Clarifying to what extent maximal-entanglement EAQECCs attaining the entanglement-assisted Griesmer bound can be achieved for arbitrary $k$ beyond $k=3$ (Araya et al., 2019).
• Understanding the full range of hull dimensions possible for optimal, MDS, and almost-MDS codes, especially in the context of applications to quantum information.

A plausible implication is that, for large $q$, the abundance of MDS Hermitian self-orthogonal codes extends to the broader class of Hermitian $\ell$-complementary codes, suggesting eventual classification results for general hull dimensions and further optimal constructions for quantum codes (Wang et al., 14 Dec 2025, Sok, 2021).