Hermitian Construction Methods

• Hermitian Construction is a family of methods embedding Hermitian duality, metrics, or projections into structures across coding theory, complex geometry, and K-theory.
• It underpins practical designs, such as generating self-orthogonal, dual-containing, and LCD codes, along with constructing quantum stabilizer codes using matrix-product and unitary techniques.
• Implementations extend to forming Hermitian–Einstein metrics in complex geometry, producing orthogonal projectors in representation theory, and unifying constructions in higher K-theory.

In the literature covered here, Hermitian Construction does not denote a single universal procedure. It denotes a family of constructions in which a Hermitian inner product, Hermitian duality, Hermitian metric, or Hermitian projection is built into the defining mechanism. In coding theory, it most often means constructing classical Hermitian self-orthogonal or dual-containing codes over finite fields and then passing to quantum stabilizer codes, or enlarging such codes by matrix-product, unitary-matrix, quasi-cyclic, and LCD-preserving methods (Jitman et al., 2017, Galindo et al., 2020). In complex geometry, it denotes constructive schemes for Hermitian–Einstein metrics and Hermitian Yang–Mills instantons (Fan et al., 29 Jun 2026, Park et al., 2024). In higher-categorical $K$-theory, it denotes the hermitian $Q$-construction and its comparison with the real $S$-construction (Heine et al., 2024). In representation theory and matrix analysis, it denotes the construction of Hermitian Young projectors and Hermitian almost-companion matrices (Alcock-Zeilinger et al., 2016, Markovich et al., 2023).

Domain	Hermitian datum	Output
Coding theory	Hermitian inner product, Hermitian duality	Classical self-orthogonal, self-dual, LCD, and quantum stabilizer codes
Complex geometry	Hermitian metric, mean curvature, HYM projector	Hermitian–Einstein metrics and Hermitian Yang–Mills instantons
Higher-categorical $K$-theory	Genuine duality, hermitian $Q$-construction	Genuine $C_2$-spaces and real $K$-theory models
Representation and matrix theory	Hermitian projectors, Hermitian matrices	Orthogonal SU$(N)$ projectors and ACM realizations of polynomials

1. Coding-theoretic foundations

Over $\mathbb{F}_{q^{2m}}$, the basic Hermitian involution is $x \mapsto x^{q^m}$, and the Hermitian inner product on $Q$0 is

$Q$1

The Hermitian dual is

$Q$2

and Hermitian self-orthogonality means $Q$3 (Galindo et al., 2020). In the standard case $Q$4, this is the usual Hermitian duality over $Q$5.

A central classical mechanism is the matrix-product construction. If $Q$6 are linear codes with generator matrices $Q$7, and $Q$8, the matrix-product code

$Q$9

has length $S$0, dimension at most $S$1, and satisfies

$S$2

where $S$3 is the minimum Hamming weight of the length-$S$4 code generated by the first $S$5 rows of $S$6; equality holds for nested inputs $S$7 (Jitman et al., 2017). When $S$8 and $S$9 is nonsingular,

$K$0

with $K$1.

The Hermitian self-orthogonality criteria in this setting are notably flexible. If $K$2 is diagonal and every $K$3 is Hermitian self-orthogonal, then

$K$4

If $K$5 is anti-diagonal and $K$6 for all $K$7, then the same inclusion holds even when the inputs are not individually self-orthogonal (Jitman et al., 2017). The quasi-unitary and anti-quasi-unitary cases,

$K$8

appear as immediate corollaries, and the reversal condition $K$9 yields Hermitian self-duality.

Explicit matrices are available. For instance, when $Q$0, the Fourier/Vandermonde-type matrix

$Q$1

satisfies

$Q$2

This strict decrease in the column-distance profile improves the lower bound on $Q$3 (Jitman et al., 2017). The paper gives explicit examples such as Hermitian self-orthogonal codes with parameters $Q$4, $Q$5, $Q$6, and $Q$7.

A parallel finite-field line concerns Hermitian LCD codes over $Q$8. For a generator matrix $Q$9, the basic criterion is

$C_2$0

and more generally

$C_2$1

Every quaternary Hermitian LCD code admits a generator matrix with orthonormal rows,

$C_2$2

(Ishizuka, 2022). The same paper gives a hull-preserving transformation $C_2$3 for systematic generators $C_2$4, where isotropic and pairwise Hermitian orthogonal vectors $C_2$5 satisfy

$C_2$6

and the transformed code preserves the Hermitian hull dimension because

$C_2$7

It also establishes a sharp puncture/shorten criterion: if $C_2$8 and $C_2$9 is the $K$0-th column of $K$1, then

$K$2

(Ishizuka, 2022).

A related construction over $K$3 extends Hermitian LCD codes from $K$4 to $K$5 by adjoining two coordinates derived from Hermitian inner products with an even-weight vector $K$6. In the Hermitian case, the new Gram matrix becomes

$K$7

so Hermitian LCD is preserved (Harada, 2021).

A further self-dual branch uses unitary matrices over finite fields. If $K$8 and $K$9 is chosen so that $(N)$0 in odd characteristic, or $(N)$1 in characteristic $(N)$2, then $(N)$3 satisfies $(N)$4 or $(N)$5 respectively, and

$(N)$6

generates a Hermitian self-dual $(N)$7 code because

$(N)$8

(Sok, 2019). This framework produces many MDS or almost MDS Hermitian self-dual codes.

2. Quantum stabilizer, rank-metric, and locality constructions

In coding theory proper, the phrase Hermitian construction most often refers to the passage from Hermitian self-orthogonal or Hermitian dual-containing classical codes to quantum stabilizer codes. In the standard CRSS-type statement, if $(N)$9 is an $\mathbb{F}_{q^{2m}}$0-linear $\mathbb{F}_{q^{2m}}$1 code with

$\mathbb{F}_{q^{2m}}$2

then there exists a $\mathbb{F}_{q^{2m}}$3-ary stabilizer quantum code with parameters

$\mathbb{F}_{q^{2m}}$4

(Galindo et al., 2020). The same paper generalizes this from $\mathbb{F}_{q^{2m}}$5 to $\mathbb{F}_{q^{2m}}$6: if $\mathbb{F}_{q^{2m}}$7 is an $\mathbb{F}_{q^{2m}}$8-linear $\mathbb{F}_{q^{2m}}$9 code with $x \mapsto x^{q^m}$0, then there exists a $x \mapsto x^{q^m}$1-ary stabilizer code with parameters

$x \mapsto x^{q^m}$2

(Galindo et al., 2020). This is implemented by field restriction from $x \mapsto x^{q^m}$3 to $x \mapsto x^{q^m}$4.

The same Hermitian mechanism also appears in binary quantum codes from quaternary quasi-cyclic codes. For a 2-generator index-2 QC code over $x \mapsto x^{q^m}$5,

$x \mapsto x^{q^m}$6

the paper establishes sufficient conditions ensuring Hermitian dual containment, including

$x \mapsto x^{q^m}$7

together with $x \mapsto x^{q^m}$8 for the dimension formula

$x \mapsto x^{q^m}$9

The Hermitian construction then yields a binary stabilizer code

$Q$00

(Lu et al., 2022). This produced 30 binary quantum codes improving Grassl’s tables, including $Q$01, $Q$02, $Q$03, and $Q$04.

A different specialization concerns quantum rank-metric codes for stacked quantum memory. Here the classical input is a Gabidulin code over $Q$05 with Hermitian self-orthogonality guaranteed by a self-dual basis and the condition $Q$06. The resulting construction yields quantum rank-metric codes on an $Q$07 stacked memory with

$Q$08

and parameters

$Q$09

(Nizuka et al., 4 May 2026). The paper emphasizes that this replaces CSS orthogonality with Hermitian orthogonality and removes the old odd-by-odd square-layout restriction.

Hermitian construction also transfers locality. If $Q$10 is an $Q$11 Hermitian dual-containing classical LRC with locality $Q$12 and $Q$13, then there exists a qLRC

$Q$14

with the same locality $Q$15 (Li et al., 19 Aug 2025). Using NMDS codes that support $Q$16-designs for $Q$17, the paper constructs three explicit optimal families, including

$Q$18

with locality determined explicitly by $Q$19, and proves optimality with respect to Singleton-like bounds (Li et al., 19 Aug 2025).

A useful terminological point is that two equivalent-looking formulations coexist. Some papers start from self-orthogonal codes and write $Q$20, yielding parameters of the form $Q$21 (Galindo et al., 2020). Others start from dual-containing codes and write $Q$22, yielding parameters of the form $Q$23 (Lu et al., 2022, Li et al., 19 Aug 2025). The underlying stabilizer mechanism is the same.

3. Algebraic-geometric and finite-geometric Hermitian constructions

A distinct Hermitian construction arises from generalized Hermitian curves and Hermitian hypersurfaces over finite fields. For odd $Q$24 with $Q$25, $Q$26, and $Q$27, the generalized Hermitian curve over $Q$28 is defined by

$Q$29

The paper constructs explicit AG codes from the divisor

$Q$30

and the evaluation divisor

$Q$31

where the affine points satisfy

$Q$32

(Hu, 2015).

The key constructive step is an explicit basis of $Q$33. With

$Q$34

the monomials

$Q$35

form a basis of $Q$36, where $Q$37 is defined by explicit linear inequalities coming from the divisors of $Q$38 (Hu, 2015). This yields a generator matrix by evaluating basis elements at all rational points in $Q$39. The dual remains in the same family: $Q$40 with explicit constants

$Q$41

(Hu, 2015). A concrete example over $Q$42 has parameters

$Q$43

and the paper verifies that it oversteps the Gilbert–Varshamov bound (Hu, 2015).

The finite-geometric side studies Hermitian hypersurfaces

$Q$44

over $Q$45 with $Q$46 (Blake et al., 2014). The number of $Q$47-rational points is

$Q$48

For $Q$49 this gives $Q$50, and for $Q$51 it gives $Q$52.

The same paper gives a recursive construction of the full rational point set. Writing $Q$53 and $Q$54, one has

$Q$55

The mechanism is the norm map: if $Q$56, then

$Q$57

and the equation $Q$58 has exactly $Q$59 solutions in $Q$60 (Blake et al., 2014). The paper also derives the zeta function of diagonal hypersurfaces from Wolfmann’s formulas and observes that the recursive point-set organization is relevant to evaluation codes on Hermitian surfaces. For linear forms, the resulting code $Q$61 has exactly two nonzero weights,

$Q$62

(Blake et al., 2014).

4. Hermitian metrics, Hermitian–Einstein iteration, and HYM instantons

In complex differential geometry, Hermitian construction refers to the explicit production of canonical Hermitian metrics. For a stable holomorphic vector bundle $Q$63 over a compact Kähler or Gauduchon manifold $Q$64, the recent iterative construction defines Hermitian metrics $Q$65 by

$Q$66

The paper proves that for any initial Hermitian metric $Q$67 there exists a unique sequence $Q$68 converging smoothly to a Hermitian–Einstein metric $Q$69 satisfying

$Q$70

(Fan et al., 29 Jun 2026).

The analysis uses the tensor $Q$71 and the endomorphism $Q$72, together with the comparison endomorphism

$Q$73

A central estimate is the discrete energy dissipation inequality for the Hermitian–Yang–Mills energy

$Q$74

namely

$Q$75

The proof is independent of Donaldson’s variational framework and works on Gauduchon manifolds because the Gauduchon condition $Q$76 eliminates the torsion contribution appearing in the energy computation (Fan et al., 29 Jun 2026).

A related but distinct geometric construction concerns Hermitian Yang–Mills instantons on six-dimensional coset manifolds. Using the isomorphism $Q$77 and generalized six-dimensional ’t Hooft symbols $Q$78, the spin connection $Q$79 produces an $Q$80 gauge field

$Q$81

The key decomposition of two-forms is

$Q$82

with respect to the operator

$Q$83

where

$Q$84

The HYM curvature lies in $Q$85 and satisfies

$Q$86

(Park et al., 2024).

The paper works this out explicitly on $Q$87 and on $Q$88 cosets. For $Q$89, the HYM part satisfies

$Q$90

and the topological invariants include $Q$91 and an HYM contribution with

$Q$92

(Park et al., 2024). This use of Hermitian construction is thus not algebraic-coding-theoretic, but gauge-theoretic: the spin connection is reorganized into an HYM instanton by Hermitian projectors intrinsic to the $Q$93-structure.

5. Hermitian projectors, almost-companion matrices, and constructive Hermitization

In representation theory, Hermitian construction refers to replacing non-Hermitian algebraic projectors by orthogonal ones. For a standard Young tableau $Q$94, the ordinary Young operator $Q$95 is idempotent but generally not Hermitian. A recursive construction replaces it by

$Q$96

where $Q$97 is obtained by removing the box labeled $Q$98 (Alcock-Zeilinger et al., 2016). The resulting family satisfies

$Q$99

For lexically ordered tableaux there are compact closed forms,

$S$00

and the MOLD construction further compresses the recursion (Alcock-Zeilinger et al., 2016). The earlier Hermitian Young-operator paper gives the same objective in operator form: to project orthogonally onto irreducible SU$S$01 subspaces carried by $S$02 (Keppeler et al., 2013).

In matrix analysis, the term appears in the construction of Hermitian almost-companion matrices. For a cubic polynomial

$S$03

one first passes to the depressed cubic

$S$04

An ACM is any matrix whose characteristic polynomial equals the prescribed polynomial. In the real-coefficient cubic case, the necessary and sufficient condition for a Hermitian ACM is

$S$05

and the Hermitian ACM can then be written as

$S$06

with

$S$07

(Markovich et al., 2023). The same paper characterizes unitary ACMs of degree $S$08 by an explicit coefficient parametrization and a concrete unitary matrix $S$09.

A closely related constructive program is Hermitization of non-Hermitian Hamiltonians. On discretized quantum graphs, one solves

$S$10

for a positive metric $S$11 or an indefinite pseudometric $S$12. For the six-site toy model, the paper gives an explicit sparse $S$13 and a diagonal positive metric $S$14 in the symmetric case $S$15; more generally,

$S$16

remains positive under an explicit inequality on $S$17 (Znojil, 2011). In finite-dimensional PT-symmetric quantum mechanics, Hermitian matrices arise as the special case of PT-symmetric matrices for which the dynamic metric reduces to the identity; the paper makes this explicit in the $S$18 and $S$19 parameterizations (Wang et al., 2010). These constructions are not usually labeled “Hermitian construction” in the narrow coding-theoretic sense, but they are constructive Hermitian replacements of the same formal type.

6. The hermitian $S$20-construction and real $S$21-theory

In higher algebra, Hermitian construction refers to the hermitian $S$22-construction for exact or Waldhausen $S$23-categories with duality. For an exact $S$24-category $S$25, the simplicial object

$S$26

is defined by twisted-arrow diagrams satisfying exact-square conditions, fibrancy in one direction, and cofibrancy in the other (Heine et al., 2024). Degree $S$27 identifies canonically with the exact-square category: $S$28

With genuine pro-duality, one obtains the hermitian upgrade

$S$29

again subject to the same exactness constraints. This makes $S$30 a Segal object in $S$31 and then in $S$32 (Heine et al., 2024). In degree $S$33 there is a canonical equivalence

$S$34

The comparison target is the real $S$35-construction $S$36, whose $S$37-simplices are exact diagrams in $S$38 with $S$39. After edgewise subdivision, both $S$40 and $S$41 are Segal objects. The main theorem states that for a Waldhausen $S$42-category with genuine duality,

$S$43

is an equivalence of simplicial exact $S$44-categories with genuine duality (Heine et al., 2024). Consequently,

$S$45

The paper also identifies the Poincaré-category formulation. Using

$S$46

the hermitian $S$47-construction of Calmès–Dotto–Harper–Horel agrees levelwise with $S$48 after translation of input data, yielding a natural equivalence of genuine $S$49-spaces (Heine et al., 2024).

Fixed points recover classical Hermitian invariants. Hermitian objects are homotopy fixed points,

$S$50

and under the standard exactness and idempotent-completeness hypotheses the $S$51-fixed points satisfy

$S$52

This places Grothendieck–Witt theory, real $S$53-theory, and the hermitian $S$54-construction inside a single genuine-equivariant framework (Heine et al., 2024).

This suggests a broad but precise unification: across coding theory, complex geometry, representation theory, matrix analysis, and higher $S$55-theory, a Hermitian construction is a procedure that replaces a merely linear, exact, or symmetric structure by one controlled by Hermitian duality, a Hermitian metric, or a Hermitian projector. What changes from field to field is not the formal role of the word Hermitian, but the ambient category in which the construction is performed.